felt much more sturdy than the two seaters I'd flown in previously (a Bulldog
I must say I am surprised that you found the Pa28 more STURDY than a Bulldog! Actually the Bulldog is far more sturdy being built for aerobatics, the Piper will however be more STABLE as it is really a touring aircraft and built for travelling from A to B with minimum interference from the pilot rather than for actual enjoyment of FLYING!
Unfortunately this is very much the case with most modern aircraft available for PPL training and there is very little to be gained by which one you learn on, so I would pick the one YOU feel best in.