Is it possible to drill with a unit other than your own in the National Guard?

Talk to your readiness NCO and CO. They'll talk to the readiness NCO and CO at the unit you want to drill at. The new unit will confirm your attendance and your old unit will pay you. I have done this in the past and as long as you're in good standing they shouldn't have any issues with this

Drilling at another unit is the same process as asking for an absence from drill or coming in to work before or after drill, must be approved by the commander.

Contact your AGR, request a SUTA or RST, whatever your state is calling it. Must be approved by your commander, if he's not getting it done, contact the commander.

When approved. they provide you with the certificate authorizing it, the AGR at your new unit would sign certifying that you performed duty and send it back to your losing unit to pay you for drill.
If you plan to do this for several months, you must request it every month, and ensure the document makes it back to your unit.
No certificate, no pay.

Yep you can do RST (rescheduled training) and both your unit commander and the unit you wish to drill with must also agree to the drill. You would then complete a 1380 for drill weekend and have someone at the unit you RST with sign your 1380 to submit.

I’ve done it several times.