More reading (I'm very bored during my enforced isolation)
It seems your symptoms (hot coil) are very common when running without a ballast resistor, and as you're taking the live feed for the Dizzy from the coil I'm pretty sure you don't have one. There is a possibility that you've unintentionally fried the Unilite.
Solutions?
Fit a ballast resistor and ballast resistor coil (needs to be the right ohm rating) - fairly cheap to do (under £50). There may be an issue with starting due to voltage drop, but unlikely at the dizzy is electronic and has 12V power. This will also be dependant on the Unilite being OK.
Fit an alternative Distributor that doesn't require a voltage drop.
