On 3730, if able to read a voltage or ohms, you are probably closing backplane relay #1 to route a closed matrix crosspoint on row 1 to the internal voltage meter of 3706A.
Amps measure input terminals on a multimeter are different from terminals for the VDC or Ohms.
If you want to route a closed channel to the Amps meassure, make use of backplane relay 3 and do the measurments on that row.
But you need to get the Amp meter connected to backplane relay 3.
On the rear panel of 3706A is the DB15 connector which offers you connections to the DMM terminals and the backplane relays,
Jumper the Amps HI and LO to backplane relay HI and LO on the DB15.
Now when you close a matrix cross point and the backplane, the amp meter will be in series.
NOTE: when not routed through the amp meter, the circuit will be open and no current will flow.
If that is a problem, consider placing a current completion shorting jumper on backplane relay 4 and close your current carry channels on that row.
But keep in mind max current on the switch if multiple channels will be closed to this one jumper.