Electrical resistance increases with heat. But components expand when heated. Tried checking the connectors for corrossion, cleaning them ? Hopefully not a short anywhere but that's another possibility. Typically see those types of intermittent 'works when heated' shorts in computer electronics vs actualy wires. Would be wishful thinking to hope that it was something as minor/easy as a timing belt slip.
Unfortunately there are a ton of crap, total-junk parts out there for Escorts. They claim "meet or exceed OEM standards" but that's total bull. I've gotten then from AutoZone & Parts+ suppliers. Often way more expensive, but CarQuest has been batting 1000 so far. (only drawback is they operate on banker's hours and have relative few locations) O'Reilley has only been in our area for a short time but so far they've been decent except for their "off the shelf" parts that attendants don't have to fetch. Those do suck.