I don't know anything about unilites but I would be tempted to swap it out just to change the problem. Either just slap a breaker points dist in there, or find a Mopar stock breakerless dist and look up the HEI module conversion. It is not very hard.
Unilite module is no spark and doesn't draw any current any more. Cap rotor toast. So even though Unilite ran fine for 15 years trouble free do I replace module/cap/rotor or switch to something else?
I will say this: seen many used Unilite distributors for sale on this site and have not seen any of them sell. So if I go with something else its has zero resale value from what I see no one wants them ...