Will there be any voltage drop when connecting load to batteries?
Yes, the voltage will drop.  A reasonable approximation would be the internal resistance of the cells and other resistances in the wires, switches, etc, times the current consumption.
On an unrelated note, using 10 AAA cells sounds like a bad idea; AAAs have very poor energy density (wasted space due to the casing, etc) compared to AAs.  I would suggest using larger cells and a step-up regulator.  On a related note, larger cells also have lower internal resistances.
Energizer Datasheets/Whitepapers:

Battery Internal Resistance technical bulletin
E92 cell datasheet (AAA)