35 Cabriolet distributor

[Capt Kirk]

Hello, new guy here. Before passing, my father handed down to me his 35 Ford Cabriolet. It was his car when he was a kid. Became the daily driver after he married my mom and had us kids. When I got it, it was 99% complete and in need of a restoration. My dad had the engine rebuilt and had a new top made for it just before sending the car to me. I was able to restore the car and send him a picture of it winning a trophy at the local car show...he passed 2 days after getting the picture.
I am a competent mechanic but have little experience with flatheads and could use some insight and advice...so here I am. The problem I'm having...or think I'm having...is that I believe the car has the wrong distributor on it. Years ago, my dad converted the car to a 59 A-B engine which I think is supposed to have the distributor that has a single cap facing forward. My car has the helmet distributor with a cap on each side and the coil mounted to the top. The car has very few miles on it since I completed the restoration however, during these miles it seemed to run hot and when shutting it down and restarting it sounded to me like the timing was off...a hot restart is difficult. It starts, idles and runs great when it's cold though. The timing screw is all the way down to the lower limit of the slow. It seems to run best adjusted like that but it still doesn't seem like it's enough. I'd like to keep this distributor so I don't have to remote mount a coil and it would keep some of the original nostalgia of the original 35 engine. I'd like to ask for advice and options regarding what I can do to get the car running like it's supposed to. Thanks in advance. Kirk