Why do the voltage measured across a DC motor drop when its turned on
When the DC supply sags with load current , this is a load regulation error. 

Vout=Voc-I*R
 (Voc-Vout)/Voc * 100% = Load Regulation Error
(Voc-Vout)/I = effective source impedance.

R  is the total resistance of your battery or supply and drivers.

Generally if you need 10% supply stability the source impedance must be &lt; 10% of the DC load resistance.

I suspect your supply is unregulated.

Motor RPM is proportional to V.out with no load. 
Current = torque + losses

You can increase the motor voltage easily 10% as it is the average power that affects heat rise and normally you want to operate less than the rated power.