Ford Taurus misfiring after start-up after not driving for 2 or more days

Hi all - posted here before and got some great advice, so figured I’d try again.

I have a 2001 Ford Taurus with abt 90k miles. My problem right now is that, if I don’t drive my car for a day or so, it runs rough for awhile after I start it up. The longer I leave the car without driving it, the worse things are. What happens is I start the engine, which starts fine but runs rough. Once I put the car in reverse, the engine feels like it starts running even rougher. At its worst, it runs rough enough to shake the entire car and won’t move when I press on the gas. If I wait long enough, things improve enough to be able to drive. The first couple times this happened, I was able to get the car moving and running fine after the first minute or so of driving (not idling rough or misfiring anymore). Now after I drive for a couple minutes, the engine still feels rough (very slightly), it doesn’t feel like I’m getting as much power off of the engine, and the car “jumps” a little during driving.

I recently had some issues with the coolant system (didn’t change coolant in forever and gunked everything up). In the process, I replaced the belt tensioner, idler pulley, water pump, and thermostat. Also changed my EGFR sensor for unrelated reasons. Right after that one of my engine mounts on top of the engine cracked (part was called a “torque strut”). Had the mechanic replace that part and started experiencing my current problems a week or two after that.

Car isn’t overheating or burning coolant now (from what I can tell). Oil looks good - not frothy, discolored, etc (about 1500 miles away from my next oil change). Spark plugs and wires were changed approx. 10k miles ago.

Planning on taking this to a mechanic when I have the time (overworked/underpaid grad student here!), but wanted to see if anyone had an opinion on it first. Don’t know if this could be something common or not.