if your unit has a combination throttle/choke, make sure you pull lever/control back off "choke" position quickly (as soon as engine will run w/o choking). Failure to do so - particularly on a warm engine - will flood the engine and prevent a start.
An engine still warm from previously running usually doesn't need to be choked during starting.

Are you seeing any signs of black smoke out the exhaust?
Have you looked to make sure the choke cable is adjusted properly? It's possible the choke cable has slipped since the initial adjustment made after you rebuilt the carb and now it no longer opens fully. That condition could make it easy to start when stone cold but a real bear to start when warm, and with heavy fumes. You should also see some black smoke, or blackening of surfaces near the exhaust/muffler outlets.

The opposed twin engine probably has a separate choke control. With the air filter removed, examine the position of the choke when the choke knob is pushed all the way in. The choke plate should be full open.

Your cold starting technique is probably ok - since the engine starts and runs. If the engine is warmed up, it may not require any choke to restart. If you suspect flooding, put the control to full throttle with no choke.

The most likely cause of flooding on this is a leaking needle/seat or overchoking.

This was a very good engine that has been out of production for about 10 years. To obtain longer life, pull all the sheetmetal/baffles and clean the cooling fins.