I’ve read in the past that Linux Plex servers run into a lot of issues since it was designed to run on Windows.
Very untrue. I would love to see where you read that! I've been running Plex on Linux for years now w/o incident, along with many others.
I’m not averse to buying yet another computer, but, before I do that I thought I would seek some advice if I should combine everything into one Linux server, or leave Plex as its own Windows server and put everything else on a new device.
If you're willing to learn a little bit about Docker, running Plex and all your other stuff in containers on a Linux host is definitely the best option, IMO.