Recommendations and help needed

So, here’s what my current rig looks like:
CPU: Intel Core i5 with Corsair RAM
RAM: 2 x Kingston 4GB DDR3
GPU: NVIDIA GeForce equivalent
Motherboard: MSI motherboard
SSD: Seagate 120
PSU: 450W, though I’m not actually sure which brand it is
Here's the deal: before I swapped out my motherboard, I was running an NVIDIA GeForce equivalent, but on this new board, it just wouldn't cooperate. It was super finicky—the only way I could get it to work was by toggling the power switch on and off a bunch of times until it finally decided to boot up normally. I even tested the card in another PC, and it worked perfectly fine there, so I know the card itself isn't dead. Since I had to deal with that headache, I ended up grabbing this other GPU listed above. It boots up without any issues, but honestly, it's pretty weak. I'm trying to figure out what kind of card would actually play nice with these specific components, or if anyone has any clue why my old card was acting so weird with the new motherboard. Everything else in the build stayed the same; the only thing that changed was the motherboard, which then forced me to swap the GPU.
I don't really game much; I mostly just spend my time browsing the web. If anyone has any suggestions, please let me know!
Thanks for any help you guys can give.

First off, you gotta figure out what kind of power supply we're actually dealing with. What's the exact model? How old is it? You need those details first...
Then, check the specific graphics card model. Just saying it's an NVIDIA GeForce equivalent isn't enough—you need the actual series, not just a vague category...
I mean, if you aren't even playing games (do you even play them at all?), then honestly, any basic graphics card will do fine...

Alright, I managed to dig up some info. My power supply says EVGA 500W—I messed up earlier and thought it was just 500W—and the graphics card used to be an NVIDIA GeForce equivalent. I’m looking for something with maybe up to 1GB of VRAM, though honestly, it doesn't matter much to me. Even when I play games, it's nothing intense, just stuff like chess or card games. The weird thing is, that old graphics card worked fine on my previous motherboard, but it's being totally stubborn with this new one. It actually started working after I cycled the power a few times, and it worked perfectly in my other rig, even though that PC is actually weaker than this one. As for the setup I'm running right now, the NVIDIA GeForce equivalent works okay, but it's definitely on its last legs.

You're probably running that Corsair equivalent power supply, which is honestly total garbage based on old designs. If that NVIDIA GeForce equivalent was working fine on your old setup (just to clarify: the 7000 series is the whole family, while the 7700 is just a specific line, and something like a GTX 1050 would be a specific model—see how the numbers get more specific?), it’s likely because the old motherboard could handle the power draw through the PCIe slot alone. It probably didn't even need an extra power connector since it pulled everything straight from the board (I'm just guessing here since I don't know your exact model).
From what I can tell, your current power supply doesn't even have the right cables to support a decent card...

So, when you say that GTX 1050 "works but is dying," it sounds like you haven't even sorted out the extra power requirements either... I can only imagine.
Look, if you just want to play Solitaire or Poker, any old GPU will do, or even integrated graphics (which you don't have). Your CPU isn't exactly top-tier either, so really, it all comes down to what you actually want to do with this thing and what your budget looks like...
If it were me? I'd swap out that power supply first. You can't build a reliable PC on a shaky foundation like that.

Honestly, I’m pretty sure the power supply is total garbage since this was one of those pre-built setups you just grab off the shelf. That old GTX 1050 didn't even need an extra power connector, but this GTX 1050 definitely does, and now I'm stuck. I don't really have the budget to just go out and buy random graphics cards one after another to see which ones actually work; I can't exactly play trial and error with my cash like that. Also, this GTX 1050 has started acting up—it's overheating, and even though I applied fresh thermal paste and the fan is spinning fine, I keep getting these vertical lines on the screen. I'm pretty sure the card is toast.

Your setup is great for light tasks and browsing, though that power supply looks a bit sketchy.

If the card doesn't need an extra power connector, you're likely looking at a GTX 1050 that pulls barely 50W under load (which is almost three times less than a GTX 1050). So, even that PSU should probably be fine.
Just as a precaution, have you tried plugging the graphics card into a different PCIe slot on the motherboard?

I actually spotted an EVGA 500W on Craigslist for $100 with a warranty, located in Chicago. Just in case you end up needing one...

You're totally right, my bad—it was actually an HD 7750. I tried swapping it into a different PCIe slot on the motherboard, but the exact same thing kept happening. That’s why I decided to test it with this other card instead, and that one boots up just fine, though it still acts a little glitchy sometimes. I'll get these random vertical lines popping up, or the screen just goes completely white like there's no signal at all. But honestly, once I just give the PC a quick restart, everything goes back to being perfectly fine.