Richie, 
I bought about a
hundred GE
5751s a year or so
ago and found a
number of them
were, in fact, as
you suggested, a
bit hummy!! 

Changing the bias
resistor sould not
change the hum
much at all. I think
some of those
tubes are a bit
noisy and also just
hum. 
For what it's
worth Richie, my
"loose" rule of
thumb with these
tubes is to use
about
1200-1800ohms,
under 250V on the
plate and 5uF to
10uF bypass cap
across the cathode
resistor. 
Here is a bypass
cap formula my
old mentor gave
me too many years
ago... 

1/628(.2)Rk = Ck.

This is a
reciprocal
equation and I
don't know how to
type it in any other
way.... you know
... 
one over...etc. 
It works real good
for almost all tube
guitar amp
circuits. 

Bruce