...If we assume that each side of a 20 meter dipole cut to be resonant at 14.225 should be about 16.5 feet and each side of a 10 meter dipole should be about 8.5 feet, by running them together through the eyebolt and soldering them together with the lead wire of the balun, am I creating a "single wire" of 25 feet which would no longer be resonant at the desired frequency? ...
No, they act like two separate wires as long as they are only joined at the feedpoint.
Well, actually there will be a little interaction between them, so the resonant frequencies
might not match exactly (you almost always have to do a bit of pruning in the real world
anyway, even with a single dipole wire) but close enough.
I have often put up 4 wires in parallel on each side of the feedpoint for 80, 40 (+ 15), 20 and 10m.
It does indeed work.