Ford Windstar vehicles vibrated as if they would shut down

I have two Ford Windstar minivans (2000 and 2003 model). From time to time the 2003 vehicle tend to shake if the engine is about to shut down. This typically occurs within 5 minutes after the engine is started. The problem may be alleviated (but no eliminated) if I step on the accelerator.

Similarly, the 2000 model also shakes occasionally right after the engine is started. Is this a fuel system problem (e.g., clogged fuel filter), or an air-system problem (dirty air filter?, or both?

Any suggestion on how fix it would be greatly appreciated. Tks.

When were the fuel filters and air filters last changed?
If the answer is either “I don’t recall”, or “More than 3 yrs/30k miles ago”, then you need to replace them.

Even if this doesn’t alleviate these symptoms, the filters do need to be replaced at least every 3 yrs/30k miles, lest you cause other problems.