Dodge Dakota 2001 gets stuck in first or reverse gear, then stalls

Check the slave cylinder, and specifically the fluid in the clutch master reservoir.

It may be leaking, or simply low on fluid. A bad slave cylinder (or even master cylinder) piston seal may be allowing fluid to escape, thus letting finger spring pressure from the pressure plate to engage the clutch early.

The gearshift vibration is likely insufficient or creeping throwout, which allows the clutch disk to chatter against the flywheel.

If the slave cylinder is going bad it will also probably leak. If the master cylinder is going bad, it could also create the symptoms you describe, but there should be no visible leaks anywhere in the system.

Also, if there are any general leaks in any part of the clutch hydraulics, this will result in a visible low fluid level in the reservoir.

It's quite possible it's the same fluid for the last 17 or 18 years, and although that service is often overlooked, it should be changed and topped up (if necessary - although it shouldn't be) perhaps every 20-30K miles. While it doesn't take anywhere near the punishment brake fluid does (and it is brake fluid), it is still hygroscopic, gets dirty, and should be changed at regular intervals.

The reservoir is tiny as you can see in the picture below, and it doesn't take much fluid loss to render the system useless. Fortunately your slave cylinder is external, and the entire system can be bought from aftermarket outfits such as Dorman. (Which is an excellent indication that it's a common failure and many many people need to replace this, otherwise Dorman would not market the part!)