The switches are actually jumper cables that I will connect to a PWM adc.
However, currently im connecting the ADJ pin to ground to get 1.25V and I do get it.
Yes, that would work fine. When grounding the adjust pin, you are basically turning the LM317 into an "LM7801.25" 1.25V fixed regulator.
When you disconnect the adjust pin, I'm not sure where that pin is going to be pinned and/or if it will just float. I suspect the output would just jump to ~Vin -1.5V, but you're the one with the circuit and a multimeter. That would be drawing 1.7A with your Vin and your 10R load resistor. That's 2.55W, and a TO-220 can't do that even with a small heatsink, at least not for more than something on the order of seconds or minutes.
Just as bad when you GROUND the pin and get Vout of 1.25V. That will pull only 125mA through the load. But that will drop 18.25V through the regulator. Which results in 2.25W. Almost the same. *edit, sorry, seeing your Vin is only 15V, so adjust accordingly.
If you go by the max current of 1.5A for this part in a TO-220 package at 70C using the minimum voltage drop of 1.5V, that puts max dissipation for this package at 2.25W. And that will only work as long as the part is kept below 70C, or so. And unless using a very good heatsink and a fan, you probably won't get that. I'd say with a dinky heatsink, you ought to reduce that by 50%.
You should have input and output caps, yeah, but I doubt oscillation will be causing the device to die, since your load is just a dummy resistor. Personally, I have never seen an LM317 perform a thermal shutdown. At least not in a way that it continues to work after it cools down. But then I'm in the same boat as you. I bought mine off eBay many years ago, and I can't be sure they're real.
FWIW, my own LM317 bench supply has a large aluminum heatsink with integral fan and a 20V input like yours. I have never really documented the limits, but a 10 ohm load would be pushing it at any Vout.