I assume that the two monopoles are pointing in opposite directions to make the dipole.
If they are pointing the same direction to form a parallel transmission line stub, then your
VNA results might make sense.
The connection to the transformer appears to be correct, though as KK5J noted you could
try leaving the center tap on the secondary open to see if that changes things. But if
you don't see anything
changing around 500 MHz on the VNA then I'd suggest
something major is wrong: a shorted connection, or the wrong length elements (total
length of the dipole should be about 11" or 285mm.)
You might start by connecting the dipole directly to the coax without a balun
to see what result you get on the VNA: it might not be perfect, but you should see
some indication of near resonance.
There are many other types of balun that are practical at UHF. The one with a second
length of coax parallel to the first, with the shields connected together 1/4 wave back
from the feedpoint, is common: the second conductor does NOT need to be coax; it
can be any conductor about the same diameter. I've also seen this built using brass
or copper tubing, which is stiff enough to support the dipole elements attached to
the ends. To do this, build a "U" or shorted stub 1/4 wave long and attach the dipole
elements at right angles on each end, pointing in opposite directions. Then the
coax can run up inside one of the tubes to the feedpoint. (There the shield connects
to the end of the tube with the coax inside, and the center conductor reaches over
and connects to the other side of the stub and dipole.) Do not
apply the velocity
factor corrections to the coax: that applies inside the coax, but the velocity factor
of the quarter wave stub in this case depends on the OUTSIDE of the coax. By using
some air spacing the velocity factor will be close enough to 1.0 that you don't need
to make any correction for it.
Another way to make a balun is to use a quarter wave of brass or copper tubing with
the antenna end open and the far end shorted to the coax shield: that also forms an
open (coaxial) quarter wave stub that looks like a high impedance to common mode
Other approaches might work depending on your application:
The first 70cm antenna that I worked with used a conventional half wave coaxial
balun (4 : 1 voltage balun) with a delta match to a continuous driven element: the
tap points on the antenna were adjusted to get a good match.
A half wave coax balun can also be used with a folded dipole.
G0KSC has a page on balun methods and construction here: http://www.g0ksc.co.uk/creatingabalun.html