From Sailboat Data, it seems the US 305 has a large beam-to-length ratio, long and shallow 'fin'
keel set well aft, small spade
rudder without skeg or fairing of any kind to direct
water flow over it, plus a very large overlap on the
genoa. All these combined will likely cause a less than desirable amount of control so I'm not surprised she does not handle like your previous
boat (Catalina). She simply does not look balanced. To answer your question: it's almost certainly the
boat, not you.
That small, shallow spade
rudder may well simply 'stall out' if put hard over to
head downwind with the large
genoa set. ["Stall", as in an aircraft wing 'stalling' and losing all
lift when the 'angle of attack' to the airflow is too great, hence the need for 'flaps' along the leading edge during take-off and landing - somewhat equivalent to the effect of a skeg on a sailboat rudder.] The same might happen if hard pressed running downwind, like many yachts that have a tendency to broach uncontrollably when their rudder loses its 'grip' on the water-flow over it.
What to do? As several folks mentioned, the effect on
steering of a large overlapping genoa is counter-intuitive. You might
experiment with the genoa roller-reefed down to a more manageable size, keeping the leech well forward of the
mast until you learn much more about her behaviour under sail. You may eventually want to permanently reduce the size of that sail. I'd be tempted to
experiment with a
cutter (or 'slutter') rig but that's for later. Reefing the main early and keeping her upright should also help. Experimenting with
mast rake may offer further improvement.
It's not easy to increase the size of rudder as all of the supporting
hull and structure around the rudder shaft would likely need beefing up as well - maybe easier to add a small skeg, if that turned out to be necessary, as the rudder doesn't seem to have much (or any) 'balance' ahead of the shaft. (I tried to attach a thumbnail from Sailboat Data showing the unusual
hull profile but stuffed it up, sorry.)
Windage of hull is not the issue, as the high bow and aft-of-centre
keel should assist in turning downwind - try it and see if she tends to fall off the
wind when motoring.