The MOST likely cause is the seat switch.

Make sure it is plugged in and closed when occupied.
Another possibility is a bad Operator Presence Relay.
Check the connections.
IF it isn't getting power from the seat switch, it's in a "kill spark" condition.

With the key ON, opening/closing the seat switch should cause the OPR to CLICK.

Could also be that it's running on only one cylinder and when the PTO is engaged the engine dies.

We tried but it only starts when I pull out the choke. It runs fine until I try to turn on the blades, it shuts off and it smells strongly of gas. At first I thought it was the battery because it wouldn't start. I replaced it and now it will start, but seems to flood out when I increase speed or try to turn on blades/mowing deck.

JAL may be on to something.

Disconnect each spark plug wire (ground them) alternately and see if you have a dead cylinder.
Also check for spark from each side.

Possibly a blown head gasket? but we can get into checking that once you verify a weak/dead cylinder or not.

This engine should be a Kohler CV730-0017.

Lost spark on one side or could have popped a pushrod...?

Thanks...I will check these things out and let you know....