What happened with your FM antenna search? It is nice to follow up on recommendations, esp something like a cheap FM dipole.
It seems the longer stays on the worse it gets. Starts out barely listenable and by about twenty minutes or so it's a garbled mess. Could this mean a heat-related issue?
Yes something is faulty. Check the power supply to the tuner first. it should be 13.5V. It could also be dirty switch contacts. Has this units switches been thoroughly clean with de-oxit D5. Usually the first line of defense. Read up on how others manage this task on sx-x50 units, they share many common problems and solutions.
Many of these old components show tarnish, it does not necessarily mean that they are faulty. testing tells you if they are faulty, not always looks.
My sx-950 has a cover over the RF front end. I have seen others that do not have one. It does not mean that it is necessary for the tuner to work properly.
When you are looking for a DMM, for audio work, look for accuracy, AC performance, like what is the AC frequency response, that way it can be used to measure audio freq signals through the range up to at least 20KHz. I use old test equipment such as a HP 3478A. Problem with old equipment is that it can fail at any time and if you are not up to fixing them then they can be useless.
You also need a sine wave source in order to provide a stimulus to be able to trace a signal thru the ckts.
Investing in test equipment can be a expensive and if you only use it once it can be a waste of money.