Check valve clearances.The backfire and the sluggish running could be due to an exaust valve(s) hanging open.It's easy and if you do it yourself it's free.Before shelling out $$$ for carb work.
Spec is .002" to .005".Try this,turn off the kill switch and crank the engine.Does it sound like all 4 cylinders are building compression?If they sound O.K. take it for a ride and check it again hot.
If the base engine isn't right the carbs don't even matter.And check compression too while you've got the tank off a burnt valve will cause what you're describing too.
Good luck!
liljohnnie69