1325 Movement advice

hello all,
I bought a vintage omega DeVille with a quartz cal1325 movement which has two problems. for some reference on me, I'm not watch-repair-inept and I've disassembled movements before and returned them to proper order, though not quartz.

The first, is that it intermittently stops and starts. After cleaning the battery compartment and a few battery changes I've had mixed results with keeping it going, but almost always it will lose 15 minutes or so before continuing on(sometimes).

So I assume theres gunk, old oil, maybe a broken gear, maybe something else in the works that is causing the issues. I assumed I could remove the hands face and dial and attempt to clean the movement with “quick clean quartz movement spray” but I'm wondering if a movement swap isnt just the better option, and if so, are there 1325 quartz movements available somewhere besides looking for a donor watch.

The second is that the stem has randomly fallen out and I cant seem to get it back seated. I assume theres a trick to getting it back in on this, but I havent been able to find it.

any tips?

You could try oiling with some 9010, if you are taking the hands off, remove the dial plus oil those pivots.

Obviously it may work or not, that accumulative dirt restricts function.

Good luck.

any ideas for the stem not going back in?