Vintage Integrated Amplifiers With Great Sound

I could budget up to $3500 for something really nice. 

I still don't know what loudspeakers you are driving but let me throw this out there. A Musical Fidelity integrated. They will match well with most loudspeakers and have a classic appearance. I would try to find one that is less than 20 years old to avoid a lot of potential problems. There is a really affordable 3.2 on sale right here. 

Yamaha went retro with their newer amps. This A-S1200 looks like a vintage amp and it also has power meters!!

I agree that Musical Fidelity would be a good choice. If you can find an A308, you’d be getting the cream of the vintage crop. Check out reviews.

Vintage made in England Musical Fidelity integrated are great values. I purchased a A220 a few months ago for $650. I upgraded all caps to first tier and it sounds sweet through my KEF speakers. 

Vintage integrated have a problem with digital. You'd need to budget for a DAC and interconnects, and then it's not integrated anymore. It wasn't until Flac came out (2001), and then got mainstream support (lets say about 5 years later), that digital even started getting good, so 2006ish. Prior to that, there really just aren't any good integrated that handle digital. Newer DAC chips are better too. 

If you don't use digital then fine, but digital makes a lot of sources these days, and it's really convenient. Newer integrated amps will just support it better.

For example, a Synthesis Roma 96DC+ is a really strong contender for an integrated tube amp and you can find them used in your price range. Get a network streamer and use the Coaxial SPDIF input and if you already have some sensitive speakers then boom, done. Or the non "+" version if you want phono. Frankly I like Synthesis better than McIntosh.

But that's just like, my opinion man. Try to find a place you can hear some more things. Luxman and Primaluna may be good options too.