OP wrote:
"Was this really an option to you? I have just 8 gb of RAM in total, so when I remove one of the slots it's really slow."
Just to be clear, you have a 2012 non-retina MBPro with removable RAM and drive, is this correct?
Could it be -possible- that when you put a little pressure on the back cover (by picking it up), contact is being made with the RAM DIMM? Not much "deflection" would be necessary to make things go awry.
I've been seeing a number of posts from folks who have non-retina MBP's, and have "memory slots go bad" on them. I'm wondering if this could be a problem with "movement" or "flexion" of the back, pushing on the RAM DIMMs? (I could be completely off-track here).
Here's what I'd suggest for you:
First, remove the "topmost" DIMM (the one that is "closest to the cover").
Even if it's slow for now, close it back up and run that way for a day or two. Do the screen errors go away after you do this?
If so, I would consider doing this:
1. Buy ONE 8gb DIMM, and put it into the LOWER slot. I would leave the topmost slot EMPTY. That should give you enough RAM to do most things.
2. Do you still have the original platter-based hard drive inside? If so, REPLACE IT with an SSD. Things will IMPROVE GREATLY (I cannot overemphasize this). It's easy to do (since you already have had the back off, it's child's play).
IF running with the single (lower) RAM DIMM solves the screen glitches, the SSD will breathe new life into the MBPro and remake it into a good performer for the next 2-3 years.
Additional EDIT:
The 2012 MBPro 13" models also have a problematic hard drive ribbon cable. If the cable starts to go, communication is lost momentarily between the drive and the motherboard, and it will mimic the symptoms of a bad OS or failed drive (but it's just the cable).
Apple (for a time) had a free replacement for this cable. Not sure if that's still in effect.
Do you have a brick n mortar Apple Store near you?
If so, make an appointment with the genius bar, and let them check the cable for you (perhaps test the RAM at the same time).