on the main motor, checking the main windings (P3, P4, &P5) these are the Main Winding, Aux. Winding-Wash, and Aux. Winding-Drain. When I check across these connections, I have continuity to each and all of these three connections. Is this how it is supposed to be.

Yes

They are all connected together at the connection on the right hand side of the motor on the wiring diagram (R-BK). To check each one individually one meter lead must be on the R-BK and then the other lead to P3 or P5 or P4.

As an example of what you are doing if Main winding is 3 ohms and Wash winding is 5 ohms, when you measure P3 (Main) to P4 (Wash) you are measuring those 2 windings in series so you should read 8 ohms.