The oil inside the distributor could cause problems. There is a seal that is possible to replace, but it is not straight forward. The Honda dealership doesn't sell the seal and make you buy the distributor housing. Send TexasHonda a message, because he has a procedure to change it. There is also information on where to get that seal.

Your problem sounds more like a transmission or transmission computer problem. I am not good at giving advice on how to test and verify a transmission issue.

 

Could get that seal from a junkyard or ebay.
Part #6 all the way to the left I presume.

 

Part #6 is the O-ring around the base of the distributor. Leaking there would run down the side of the head, NOT INSIDE the distributor.

Oil inside the distributor is from a leaking seal where the shaft spins inside the distributor body. You won't find that in the FSM or parts counter because Honda's official story is that the entire distributor is replaced as a unit (part #4).