Like already reported, the FA-5650A is a noisy devices; first reports about its uselessness for microwave applications date back to about 2010 from KA7OEI, to my memory he also shows a way how to clean up the signal with a rubidium locked oscillator
Interesting, initially I'd thought the "FA-5650A" was a typo for the "FE-5680A" but when I googled " FA-5650A" to check, the only results led to a few hits for the FE-5680A and this topic thread. However, googling for FE-5680A led me to results for both model numbers with one of the results being this informative page :-
https://prc68.com/I/FEIFS.shtml.
Both models use the same DDS method so need the use a 'clean up' VCXO solution to allow them to be used as 10MHz reference frequency sources for GHz rated transverters (the Efratom LPRO-101 and its variants can be used directly without the need for a 'clean up' add on VCXO).
Personally speaking, the Efratom LPROs are a safer purchasing choice imo since you can never be certain which of the '57 varieties' of the FEI models you might actually land up getting lumbered with (the issue of remaining lamp life and whether you get a working unit applies to all of these used RFS models regardless of make and model).
The bonus features of 1PPS and, where enabled, programmable frequency setting via a serial interface, are rather overrated imho since the 1PPS has no fixed relationship to the GPS PPS time reference pulse anyway and it makes very little sense to use the RFS directly as a variable (and limited range) frequency source when it's far better to simply employ it as a fixed 10MHz reference for test and measurement kit and use a separate RF generator or AWG to generate a much less limited range of frequencies locked to your 'Atomic reference'.
The Efratom LPRO models are more or less a "Straight out of the box" 10MHz secondary atomic reference solution whereas the FEI models might involve some modification or reprogramming to achieve this happy state. If you're prepared to add a clean up oscillator, you can phase lock this to whatever oddball frequency your unit happens to produce if it turns out not be a 10MHz unit, thus neatly avoiding having to open it up to modify it internally for 10MHz.
Adding a clean up oscillator to an FA-5650A or FE-5680A is a relatively trivial task in the grand scheme of things so, as long as any of these can produce a stable frequency, locked to the Rubidium physics package whatever oddball frequency they happen to output (8.333333MHz or 15 or 20 MHz for example), phase locking the cleanup oscillator will provide an effective solution without any need to delve into its innards to effect a possibly risky modification.
If you're prepared to do the extra work of adding a clean up oscillator (a decent quality VCXO will suffice - no need to go to the expense of a VCOCXO in this case), you can turn this to your advantage to build a cost effective RFS using one of the cheaper less desirable frequency variants of the FE-5680A or else just pay the premium for an Efratom LPRO-101 for an 'easy life' and have done with it.

Having said that ("and have done with it"), if you're serious about the ultimate stability, there's a little more work involved than just shoehorning it into a barely large enough plastic or metal case (whether sealed or ventilated) as demonstrated in some of those idiotic youtube videos I've had the displeasure of watching.

When you're chasing down stabilities of the order of 10ppt or less (as I am in my quest to quantify the ionospheric errors that plague the single frequency timing GPS based GPSDO models with phase shifts on the order of 5 to 6ns pk-pk that I've observed so far with my modern re-spin of the famously high performance James Miller design using a uBlox M8T in place of the ancient Jupiter-T GPS receiver module used in the original design), you need to hold the base-plate temperature to a very tight tolerance against room temperature variation to achieve the required stability.
In my case, that means installing it into a roomy steel instrument case, mounting it onto a quarter inch thick aluminium heat spreader with a large (80mm square by 25mm deep fan cooled) CPU heatsink bolted onto it to control heat transfer rate to the walls via recirculation inside of this unvented case using PWM fan speed control mediated by a bead thermistor literally embedded into the centre of said heat spreader.
Initial testing of the effectiveness of this recirculating fan cooling system in an unvented steel enclosure of 1800 sq cm total panel area indicates this is a viable way to control the base-plate temperature with the absolute minimum of openings for a bi-colour front panel led with two on the rear panel for a 5.5mm DC jack and an SMA-F output socket.
Although the 'recommended' DC supply voltage is quoted as "24 volts", the full voltage range is given as from 19 to 32 volts (with power consumption test figures curiously covering the slightly wider range of 18 to 36 volts). After initially testing with a 24v supply (just under 2 minutes to lock), I elected to use a 19v laptop charging brick (I have at least three suitable laptop chargers to hand) which extended the time to lock to a pretty consistent 192 seconds from a room temperature in the range 20 to 25 deg C. The reduction in energy consumption after lock up is only a modest 1W in this case but 'Every little helps'

as does keeping the PSU outside of the case (improved charging brick reliability due to lower temperature operation and faster swap out of any failed units being two other obvious benefits as well as simplifying some form of battery backup I might wish to add at a later date).
My initial idea for fan speed control had been simply to turn it on and off to control the base-plate temperature since there is a rather massive thermal inertia involved. However, my solderless breadboard lash up had introduced an accidental PWM effect around the switching point (as such solderless breadboard lashups are wont to do) and the 12mV hysteresis on the temperature sensor signal disappeared, leaving it being held to a steady reading once the temperature had stabilised which inspired me to use PWM 'on purpose' as opposed to 'by accident'. I'm now at the stage of testing my PWM circuit ideas, having finally extracted those hens' teeth from out of the jaws of all those Unicorns googling efforts.
I'm aiming for a base-plate temperature of 35 deg C which should be good for a maximum room temperature of 27 to 28 degrees or so (here, in this part of the UK, it's very rare to see summertime room temperatures go above 25 to 27 degrees). I'd prefer to avoid going to a higher base-plate temperature but, if needs must, I still have the option to drill some discreet ventilation holes to enable some external cooling airflow to avoid raising the set temperature above my 35 deg target.
Unfortunately, I won't be able to verify the actual cooling requirements until I have everything mounted inside and the enclosure sealed up ready for final testing but I do at least have a contingency plan in place should a more effective cooling solution ultimately prove to be required.
If anyone is curious about why I am going to so much trouble in using a recirculation cooling solution in a sealed case rather than 'take the easy way out' with a ventilated cooling setup, it's because this, if it works as well as my initial tests indicate, is actually the most pragmatic approach in that, as well as minimising EMI leakage paths with additional routes for ingress and egress of electrical interference, I'm also avoiding unnecessary work on modifying the case as well as maximising the possibility of repurposing it should I later decide to rehouse my RFS in a better enclosure.
I hate unnecessary work and needless mutilation of an otherwise serviceable enclosure that could be used in a later project should it ever become redundant to the current project - I like to keep my options as wide open as much as I possibly can.

John