The first thing is to fully assess the extent of all the damage (you don't want to hang new fascia and soffit from rotting rafters).

Poking around with a screwdriver will reveal areas of heavy damage, but finding nothing with the poking around won't fully rule it out. At some point you will have to start demo to know the extent of the job. Contractors know this and it makes estimating not only the cost but the amount of time difficult. So at times like this where they have more work than time, getting someone to even look at it is near impossible.

Like finding a job, finding a contractor is aided greatly by networking. Ask everyone, including strangers who they know. Don't rule any lead out. Maybe someone knows a local retired guy that can walk you through it (you do the work, they drink your beer).