Clyde1820

I'm 200lbs+, and myself prefer the mid-1980s through late-1990s vintage steel-framed, "hard tail" mountain bikes. Typical example: Trek 930, 950, 970. Though, there are plenty of other examples from Diamondback, Bianchi, Ritchey, Ross, etc. Assuming you're in the USA, if you expand your search nationwide then you're likely to find several dozen available for sale at any given time, many sub-$300 and some even sub-$100. Check for listings on CraigsList, LetGo, OfferUp, etc.

Pick up a bike, possibly rebuild the wheelset (at least, get it trued and made as reliable as possible), go through the brakes, shifters and cables to ensure everything's working smoothly, then just start riding. Won't be the latest thing, but it'll probably outlast much of the "new" stuff. It'll certainly be strong, I would think strong enough for 350lbs of moderate use.

My latest bike is a mid-'90s Trek 970. I decided to revamp the whole drivetrain to 2x11spd (up from the 3x8spd it was). Am wrapping up installing the derailleurs, brakes and cabling, then it'll be done. It'll end up far more than $300, all-told, but it would have been sub-$300 if I'd just stuck to a good cleaning and replacement of the chain, cabling and brake pads (which is all it really needed in order to be functional). Very strong bike. And with the 26-inch MTB wheels that can support ~2.5" tires, it's comfortable to ride. If you find a lower-priced example, that'd free up the rest of your $1K budget for a newly-build wheelset if you prefer, which you could then have made as strong as you need it. It's one way to go.

If you prefer a new, store-bought bike, one I might suggest: The Trek Dual-Sport (DS). Had one for several years. Very strong, certainly capable of 300lbs. They do come with a suspension front shock, but they're reasonably capable. Not an outright MTB, but not a simple road bike. With larger MTB-type tires, it'll certainly cushion the load and make getting around town comfortable.