This sounds like the camera is only partially unjammed. Based on the reports here, the advance is working again, but the shutter release linkage may still be stuck in the “fire” position. Another user with the same FE symptoms found the issue was likely related to the self-timer mechanism sticking from old grease.

A simple DIY thing to try first: operate the camera repeatedly with the self-timer engaged many times in a row. The idea is to work the mechanism free if dried lubricant is causing it to hang up. If that helps, the problem may be intermittent rather than a broken part.

If it still auto-fires after winding, the camera likely needs deeper mechanical service, not just resetting the bottom-cover lever. Since used FE bodies are often inexpensive, a practical alternative is buying another FE and keeping one as a parts body if you’re comfortable working on cameras.

So: yes, you can try a limited DIY fix, but if exercising the self-timer doesn’t resolve it, proper repair or replacement is the more realistic path.