OK, voltage is OK, and remains stable assuming that you have
a hefty enough supply to keep the voltage at different loads.
Are you sinking or sourcing the LED's ?  If you have a scope,
can you look at the pins for the LED's and determine if the
dimming is caused by a current or strobe problem.  If its
current (meaning that the output does not oscillate), then
try buffering - you also mentioned that you are NOT putting
a series resistor.  You might also break the power circuit
and look at the current going to the PIC.  When you start
to exercise the part, such as during RS232 period, you might
be getting current spikes that may also be corrupting the
data.  Also check the supplies at the translator chip, or
even disconnect the translator and see how the LED's react.
Somehow, it needs to be narrowed down to hardware or software.