motor stuck or busted.

If a PSC (Permanent Split Capacitor) or a Capacitor Start motor, such as those commonly used in air handler blowers, cannot start by itself, but when rotated manually, continues to run fine, it usually indicates a problem with the capacitor. It is a rather common failure, I would say much more common than the motor bearing failure or any other mechanical failure of the motor. These AC motors do not have brushes - you don't need to worry about those.

Changing start/run capacitor is very well within the realm of DIY, as long as you get a suitable replacement. Find the original one in the air handler cabinet, not far from the motor, connected to it with wires, obviously. The run capacitor is a round or flattened aluminum cylinder with contacts on one end. When you're getting a replacement, get the same capacitance value (in mFd, micro farads),  and same or higher maximum voltage, both will be printed on the body of the capacitor. Good hardware store carry them and there are score of them available online.

<strong>Make sure to shut down the power to the unit before opening the cabinet!</strong>

Also, the old capacitor needs to be discharged because it can store a potentially dangerous charge, even when power is off. To discharge it properly you would need to connect both terminals through a 1K+ Ohm resistor and make sure you don't come in contact with either of the metal parts. However, if you don't have a resistor, many people just stick a screwdriver (holding it by an isolated handle of course) between the contacts of the capacitor. It may spark if it still holds the charge, so you need safety glasses, and the procedure is pretty bad for the capacitor. But, since you're replacing it anyhow, your safety should be a more immediate concern.

The capacitor change procedure after you discharge it, goes as follows:

1. Make a note of which color wire goes where.
2. Disconnect the wires from the capacitor.
3. Unscrew the bracket holding the old capacitor  and slide the old one out.
4. Put the new one in, tighten the screws on the bracket.
5. Connect the wires back where they were.
6. Close the air handler cabinet.
7. Restore the power and give it a go.

Good luck!

Yes, your motor require grease because it makes motor free for rotating and easy to work with power supply. The AC motor has some effective concept and it must require full fill.

The run capacitor is a circular or compressed metal cyndrical tube with connections on one end. When you're getting a alternative, get the same capacitance value.