Sorry, I misread your original question/problem statement a bit, since I didn't realize you also want to record 4K while streaming to FB at 1080p. This will be very taxing on your GPU to record at 4K.
In this instance, you'll need to set both the base & output resolution to 4K in the Video settings and then only do the resizing in the Output:Stream settings AND then in the Output:Recording settings you'll have to set the encoder to not be "(Use stream encoder)"
Could you post your OBS log here, since that should show us what version and settings your using, as well as your hardware specs (eg. CPU).
You can also see if the OBS log file analyzer shows anything useful:
https://obsproject.com/tools/analyzer
Lastly, are you playing games while doing this streaming & recording, because that one its own may warrant a high end graphics card.
I don't have experience with actual capturing at 4K yet, so I can't make any concrete suggestions on graphics cards...