It's really up to you if you want to try the PSU or not. It could work, but if it's had a lot of dust buildups and if you don't know if its stable, testing it with a multimeter might be a good way to make sure the power delivery is still as intended. If you want to do this, I would recommend you look at specific guides, since it will require tools and very specific actions. Be aware that upgrading your GPU may increase your overall system voltage and cause problems to that PSU you're using.
In the case where you decide to leave it and look for a new one, you could look at the ones I mentioned (EVGA G5 and RM750x). They will only set you back around 100$ and you could still get a GPU/SSD combo depending on what you are looking for They also both feature very long term warranty, 10 years I believe, so make sure you keep that proof of purchase to save 100$ in a few years if any complications happen! A 250GB SSD will only cost you around 60$ probably and 240$ can still get you a decent GPU.
My personnal recommendation is to either get an SSD+PSU for now and save up a bit for a new GPU (Ampere GPU's are coming soon, you might be able to grab a 3050Ti-3060 for a decent price then, if they aren't being sold out instantly, which is somewhat likely unfortunately).
As for the SSD I recommend to stick to good brands, since some "no-name brands" may sell cacheless SSD's which are as slow if not slower than conventional HDD's. Samsung 850-860 EVO's are a good option and are pretty cheap. They are my personnal favorite, I own 3 of them. For the GPU, would I recommend waiting for future releases, since GPU's are currently often overpriced and hard to get your hands on for a good price because of limited availability. Ampere cards "should" hit the market late 2020 or early 2021. Otherwise a 1660-Ti would be a great upgrade for you if you want to do it now.
If you end up buying a GPU and are having crashes under load or similar things, your PSU may be running into it's voltage limit. If this does happen, underclocking your GPU could solve the issue until you upgrade, but it's probably better to wait.
EDIT: A 1660S would be cheaper than a Ti with similar performance. You can expect over double the performance, as shown on Userbenchmark. 111% increase in overall performance and 130% average FPS increase.
https://gpu.userbenchmark.com/Compare/Nvidia-GTX-960-vs-Nvidia-GTX-1660S-Super/3165vs4056
EDIT2: I lied, I have 3 Samsung SATA SSD's and a 970 EVO Plus M.2, I guess that makes 4 of them!