I have a 50 ohm swr meter (Heathkit, Drake W4 and Bird) and I am using 70
ohm coax for a dipole antenna on 3.7 MHz.
If I trim the antenna for the lowest reflected power showing on the meters,
will the antenna be cut to resonate (or near resonate) frequency or is it
possiable the antenna will end up around 100 feet or 150 feet instead of
near a calculated frequency length of around 125 to 130 feet.

Guess the real question is can I use a 50 ohm meter with 70 ohm coax and
when the reflected power is the lowest, is the antenna system adjusted
correctly ?

I don't really worry about it, just stick up some wire and operate. Adjust
things so the rig will see a low enough swr the internal tuner will take
care of it.