This is quick mock up to help.
Imagine the break in the loop below like one of your driver leads being un-hooked from its terminal strip.
For a current measurement, wire the meter in series, but make sure to plug your probe or lead into the
10A hole, notice pic below and where the red/positive wire is plugged, farthest right hole is the 10A hole on my meter. Then select 10A on the multimeter dial.

You may have to use tape or hold the wires so they make contact, if you're wired up through your multimeter and the driver is on but no light is shining, you probably aren't making a good connection in the holes.

Once you have it in place plug the driver in and look for the current reading...
To take voltage measurement no need to break the cct. No need to un wire the terminal, just place the multimeter positive probe on the positive terminal block and place the negative multimeter probe on the negative terminal block.

Also make sure to switch the positive multimeter probe to the 200mA hole. And then also the multimeter dial has to be rotated to select the max V you think you may read. The straight line is DC, the wavy line is AC, make sure to select DC.

The driver in the mock up is rated at 60V so the 20V setting was too low, and the 200V setting will not be exceeded and be safe with only 60V, so 200V it is.
Looks like about 30V (I traced the negative wire with a bold line so you can see the cct is not broken and looping a little easier)...
I sure hope it's nothing to do with the terminal blocks, sorry you're running into issues, hopefully you get it figured out.
I think its either pulling to much from the wall on the AC side, or maybe trying to push too high V on the DC side from too many Ω's in the terminal block connections? I sure hope not but the AC side is checked easily with the kill-a-watt device that someone posted earlier, and checking DC voltage doesn't require any un hooking, so you could check those things easily. Just check what the kill-a-watt reads on each driver when they are full blast, and then test the DC voltage of both setups when the drivers are full blast, if its due to the terminal blocks I think the VDC will be higher on the terminal block setup. If this is the case you could try tightening the screws to make better connections.