EGI relay buzzing... car DOES start... kinda...

[fc3s kit]

After searching "EGI buzzing" and finding a bunch of threads where cars don't start, I couldn't quite find one that matched mine. I have a 93 KA auto. Here's the problem.

I'll try to start the car and without doubt, it will fail. The EGI relay will then buzz for several minutes. If I try to start the car while it's still buzzing, it'll still fail, but once the buzzing stops, the car will start up just fine.

I tried swapping out the relay and after 3 others, I figure that isn't the problem. I just want to know why it's buzzing and if there's anything I can do about it.

My car also has an inconsistent idle (I figure I have to clean out the IAVC), what I think is a timing guide rattle (nothing major, right?), and the sync's on 1st and 2nd might be going out (I think that explains the tranny's hesitation to get into/out of gear at low speeds, which is annoying, but nothing I can really do anything about). Just thought I'd mention those in case anything might be related.