Now that I have your attention.....when ever the shower/tub is being used,
and either a cold or hot water tap is opened elsewhere in the house, the
unlucky person in the shower/tub gets cool water only...there is some
mixture of hot and cold, but the water temp drops significantly....

The copper piping from the hot water tank feeding both washrooms is
5/8 inch (outside diameter) throughout. Now, before I go out and purchase a
couple of pressure-equalizing fixture sets and have them installed, could
increasing the hot water feeder pipes to 7/8 inch solve my problem? I would
go with cheaper, easier to install poly pipe.

Also, if my problem can only be solved with pressure-equalizing fixture
sets, are there pressure-equalizing fixtures that have two taps (hot and
cold) so that I don't have rip out tile and drywall to create one center
hole.
Thanks