Scope CRT's don't usually burn in unless you really abuse them. Don't set intensity higher than necessary to get a good trace, and don't leave it in XY mode without any inputs (it will create a single dot, which will heat up the phosphor fast). Of course phosphor does have a finite life, so don't leave it on overnight, especially with a straight line, and use the vertical position control once in a while, but the CRT isn't usually the thing that dies first. Plenty of CRT's much older than this that are still fine, although the amount of usage it saw is usually unknown.
I would start by checking the power supplies (as usual), check for correct voltage (with your DMM), they should at least be close (14.5V instead of 15V is often fine, but 8V isn't). Also check for ripple (your scope will still work as long as you only look at the first 8 div's or so, right?). Check any connectors or sockets for bad connections. Check for the already mentioned bad solder joints. Cracks in PCB tracks. Heat make things expand, so it can easily make a bad connection worse. Anything that looks burned?
Does it appear heat related, i.e. does it always occur after it's on for some time or always right after you turn it on cold? Does turning it off and on without letting it cool down for too long make the problem re-appear faster? Or is it completely random?
The main timebase and delayed timebase both have the problem, so that suggests something they have in common (eg. power supply or horizontal amplifier). It the problem went away for a few seconds when you switched from delayed to main intensified by delayed timebase, as if it had to re-stabilize or re-heat, is this consistent? If you have an idea where the problem might be, spraying cold spray (or an improvised version of that) can be informative, if the problem goes away, you've found the problem.
What happens when you change the horizontal sweep length and/or position control? Does the distortion compress/move with the control, or does it stay on the same place when the waveform moves?
The service manual should be helpful in at least localizing the problem to a certain part, and should tell you have to check the power supplies and similar stuff.