PC Problems: Upgrade PC or build new one?

Feb 9, 2019
I had my PC custom built for me a little over three years ago for school and light PC gaming/streaming. I want to be able to play and stream titles such as Fortnite and (newly released) Apex Legends.

Lately I would play these titles for about an hour or two and the game would either crash or my computer would bsod with memory management. This is without streaming. I performed a clean install of windows, updated all drivers, and checked for faulty parts. All parts checked out fine. I checked temps and my GPU is okay in temps even with 100% load but my CPU is running hot. I have not reapplied thermal paste and computer was just cleaned for dust recently. I don't know if these games are CPU intensive. Could they cause this kind of crash? Is the GPU holding me back?

- First, what parts would be the most beneficial to upgrade if I can keep some of my system? Do I overclock my CPU and aim for a better GPU? If I overclock whats a good CPU cooler to use, my Liquid Cooler doesn't seem to be working well. Might be defective because its old?

- OR second, if I need to completely upgrade my whole system, what system should I aim to build?

Current PC:
- CPU: i5-4690K 3.5 GHz
- GPU: Zotac GTX 960 4GB
- RAM: ADATA XPG V3 16GB (8GBx2) DDR3/2133MHz

- To be able to live stream my gameplay using OBS software.
- Run high intensive cpu/gpu games *(Fortnite/Apex Legends) at 1080p 144Hz if possible
- No bottlenecks

I have a tight budget, cheaper is obviously better if I can still use some of these parts, but open to suggestions of a brand new PC. Sorry for the wall of text, I appreciate all of your help.
What tests have you performed to test stability? Memtest86 bootable specifically? I've seen this myself before, and had to add a bit of DRAM voltage to correct. What are CPU temps?

Also, to get to your goals, a complete new overhaul would likely be needed. What kind of budget would you consider?