Unfortunately your transmission needs to be rebuilt. I just went through the same exact issue this past winter. Probably will cost about $1500. Good luck.
Might be just the transmission convertor lock solenoid needs replacing or a full rebuild like ColdinMA suggests.
The Trans will need overhaul most likely. Worn clutch material clogs the oil pump pickup and the car stops pulling when it gets hot... when it cools down the clutch material has settled into the bottom of the trans and the cycle starts all over again. You can TRY changing the Automatic trans fluid... BUT the damage has most likely already been done. Wish I had better news for you. :)
Looking for a Used Civic in your area?
CarGurus has 52,165 nationwide Civic listings starting at $999.
Search Honda Civic Questions
Honda Civic Experts