"On this Dayton motor T6, T7, and T8 have continuity. T7 is connected to centrifugal switch and the other non-labeled wire from the centrifugal switch is connected to the start caps. T4 and T5 are tied together to L1. T8 and T1 are tied together to L2. The run caps are tied in series with the run windings(T4,T1). Where does T6 fit into the wiring?"
First of all...
The "run caps" DO NOT go in series with the run winding, they are in series with the start winding and at the same time in parallel to the start caps and the start switch (which ARE in series).
Draw it like this.... THREE items ALL in SERIES... "start capacitors" to the "start switch" and then to the "start WINDINGS." Then the RUN CAPS are in PARALLEL with the "start caps" to the "start switch" part of the start circuit.
Following your description...
"On this Dayton motor T6, T7, and T8 have continuity" And T8 goes to L2.
OK...L2 to T8 which connects to T7 and T6.
"T7 is connected to centrifugal switch and the other non-labeled wire from the centrifugal switch is connected to the start caps."
OK...we now go thought the start caps to what I will call... POINT "X," (the other side of the start caps). Point X then HAS to be connected to the START WINDING. The start winding then FINISHES at T5, which is connect to L1.
"On this Dayton motor T6, T7, and T8 have continuity." T6 then connects to one end of the RUN CAPACITORS and from the other side of the run capacitors we connect to POINT "X."
That completes the motor you described to make it work, but I think you will find it will not work since you say "On this Dayton motor T6, T7, and T8 have continuity," and probably have not physically checked to see what those three leads are connected to internally in the motor. I suspect you will find a few ohms of resistance between T8 and T6 & T7. Till you give a more thorough explanation of how everything is connected I will leave you with what you described.