Right side hydrostatic freezes up after runing a few mins

The only thing the two hydrostatic drives have in common is a drive belt that ties them to the crankshaft. If both side quit at the same time that belt is most likely the problem.

Probably going to need new piston rings. Pretty sure your burning a little oil.

First check to see if you have any leaks especially on the problem let side. This includes the wheel motor. where hydro oil comes out, air can go in. either way it sounds like you need to take your pump out and open it up and check for metal inside. If you do not have any metal inside check the 5 pistons to see if they are stuck. What series hydro pump do you have?
10L 10A etc.

Black smoke is excess carbon, blue smoke is oil burning, white smoke is water/steam. Is your coolant level ok?

the system may have a bad condenser fan motor or it is not charged right with freon and it is over heating if the fan stays runing and it is just shuting off the compressor try to clean the coills on the ac system with water and wash them good

Sounds like alternator belt burned and broke.

- remove deck
- remove seat and rear fenders
- pry off deck adjusting knob
- remove F/R pedals
- remove floor plate
- use jackstands to prop up entire right side of tractor
- remove (2) screws holding fuel lines on side of frame
- lift out gas tank, set alongside left side of tractor
- remove plastic fan from hydrostat
- disconnect the electrical connector for the electric clutch
- from the top-side - use a 3/8" drive rachet - stick it into the tensioner slot next to the idler, rotate it to take tension off the belt, and remove belt off hydrostat pulley
- from underside - remove front idler (next to electric clutch)
- from underside - remove the "right-most" idler off the spring tensioner
- from underside - remove the "left-most" idler off the spring tensioner
- now the tensioner can rotate fully to take almost all the tension off the extension spring
- pop the spring off the tensioner arm
- the belt should now be able to be removed from the machine

- reinstall in the opposite fashion
- note: to reattach the spring, connect a nylon cord to the spring, and pull it rearward and guide it onto the tensioner arm - you need to stretch it about 1"
- for reattaching the two idler pulleys to the tensioner, get someone to hold the ratchet so the tensioner is positioned for the belt being in it's most-loose position. You may need to wrap the new belt around the hydrostat pulley first, and assemble the last idler pulley and belt at the same time to complete the procedure.

clean link from bottom of dyer and exhaust vent

That would be a head gasket leak. The motor will run fine there min. change white smoke from exhaust but min. On newer cars the check engine light will engage

The problem is that your oil and anti freeze are mixing not a good thing. Good luck .

What car , and year and engine? What color smoke?