Thank you all for your input.

I spent the morning today tracing the buried wires and found the
junction where the single wire coming from the 600W transformer is
split into different branches running to different areas of the yard.
This junction is on the other side of the driveway from where the 600W
transformer is located.

I think I have found a good place to relocate the 600W transformer so
that it will be on the same side of the driveway as where this
junction is. My plan now is to run two 8 guage wires from the
relocated 600W transformer to this junction. Then let the existing
buried 12 gauge wires run from there to each spot. It will be hard for
me to change out the 12 gauge wires after the junction, since they
cross walkways and etc. But from the transformer to this junction, I
have a clear path... which is great news!
The two 8 gauge wires will each handle 1/2 the yard. The total length
of each leg is about 40' from transformer to the junction (8 gauge),
and then another 40' from junction to each spot (12 gauge). Most of
the light fixtures are after the junction.

By the way, I used my voltage meter to measure the voltage drop across
the 2 wires coming out from the transformer. I had expected to see 12V
across them, but I dont detect a voltage drop. But somehow it
works...
How do you measure the voltage at each spot of the run if I cannot
even detect a voltage drop across the 2 wires at the transformer?
Thanks for any tips....