Try to find another 350777. You have a Vanguard series engine.
These are hard to beat. I have the same engine in a White Outdoor GT185. But I'm not willing to part with mine.
By staying with the 350777 engine, you will be assured that all your peripheral parts (muffler, carb, etc) will interchange and the electrical system parts are a match.
You did not mention why your present Vanguard needs replacing.
Inquiring minds (nosey snoops) want to know. :^)

Yep, The vanguard will run you grand or more new. Used are all that easy to find? You can check flea markets and craiglist? So, mowie guestion may be good one if can repaired it rather than replace it? Especially if you can do the work yourself?