Probably no shortage of reliable suggestions for service forthcoming. I've used Essex Camera in NJ with no complaints.

With respect, the reality is that virtually no new 35mm cameras equivalent to the FE/FM variants or F3 are currently made. I'd look at something newer like any of the Nikon AF models that work well with manual lenses. It's getting harder to find clean, low-mileage manual focus Nikon bodies that aren't all-but-dead or suffering with various issues. They're 30+ years old in some cases and their legendary reliability has practical limits--talk of pounding nails with F bodies notwithstanding.

Shop for a repair but also a replacement camera. Try to resist a DIY shutter fix. Cost? I'd guess 150-200. Your call.