It's not the transmission because (1) those symptoms are not indicitave of transmission failure and (2) the problems only started after you rebuilt the carburetor. You have off-idle stumble, which could be due to the following:

1. Accelerator pump malfunction (unlikely since you can see it dispense fuel)
2. Timing issue (possible, yet still unlikely)
3. Float height issue (very possible since this is a crucial step of a carburetor rebuild, and improper fuel level in the bowl upon a sudden increase in fuel demand will cause a lean condition)
4. Vacuum leak (very possible).