your temp sensor only gives a range of indication on the "dial," which unfortunately doesn't give accurate readings for precise temps. That could be part of the issue right there. The "pulsing" is a common issue on 1st gen explorers. There was a "bypass kit" that ford made to help mitigate the underhood "thumping" thats usually caused by a perfect storm of a few different things. Incorrect coolant/water mix (ratio) in addition to deposit buildup in the water jacket of the block, water pump potentially on the way out, or nonstandard and/or off-brand t-stat.
I'd recommend covering all of your bases first to help eliminate the culprit of your issue. Good flush of system (to include heater core,) motorcraft t-stat, repair any leaks in system, and verify you don't have a head gasket leak via fluid content test or leakdown check with a good ear. Air could be getting into the system via head crack or gasket leak. a simple bypass can be put together with some plumbing fittings and extra heater hose, if need be. if its a head and/or gasket, it'll require much more than that, obviously.
i'm sure i have forgotten some things...but the folks here are awesome on this forum. A lot of people with plenty of knowledge and experience. I'm sure they'll chime in soon with more tips. Please keep us posted as your work progresses. Cheers!