Before the German invasion, would infantry divisions be at near full on-paper strength? I seem to remember reading somewhere that they operated at about 2/3's strength. Is this correct?
For some parts of them yes.
Not all units in the Red Army were created equal in 1941. Those in the SW Military District around Kiev were in the best shape. This was the formation that Stalin was going to use to launch a preemptive strike against Germany, Hungary and Romania if he thought he needed to.
The rest were in worse shape.
Most of the infantry divisions in 1941 were very poorly equipped with artillery and had almost no motorized transport. Almost all of the motorized transport went to forming the new Mechanized Corps. This was being done in an extreme hurry since the Soviets were at least aware that there was a threat to them from the Axis.
Hope that helps.