Thank you for the advice! I have recently caved though and ordered a new one (should be here within the week).
Normally, I would agree with you and just upgrade a few things. This pc is already pretty old though. I think I did, at one point, upgrade the graphics card and later on, I added an SSD.
Problem is, it has been acting up a lot this last half year. I've had to re-install a couple of times to get rid of random freezes, reboots and crashes. Now, they have started coming back again. At one point, it refused to start at all and we had to re-seat the memory.
So I think at the very least the memory is bad, but honestly, after 6 years, I'm kind of scared other stuff will start breaking down, so I went for a new one. I'm not at that stage anymore that troubleshooting my pc excites me and gives me energy.
I was getting pretty stressed out by these things and the thought of having to dive back into the world of components did not make me happy. Making this decision has really quieted my mind!