if you didn't like kata, i don't think you will enjoy HH. main city beach is rocky, and not a swimmers beach, actually, quite disgusting lately. further south it clears a bit, but very shallow.
since you say they advertise private beach (technically illegal), but the bigger resorts, which aren't located near town, are isolated, and maintain their beachs, but you will be at the mercy of tuk tuks again, and out of town.
the seafood is rather good though. HH is a quick hop from BKK, so a lot of weekenders fill up the town. you will also have 'resorts' listed for HH and Cha Am, which seem to be in between the 2 towns, both get the same opinion from me. i enjoy HH, but only because of familiarity and a nice pit stop on the way to points south and phuket. 3 days i more than enough for me, and a pool is nice, as the beach is not swimmable, or the main beach relaxing. ugly chairs and umbrellas, horse ****/piss, as horseback walking is allowed, and extremely boring and overpriced.
if going for the 'private beach' expect a nicer beach, but isolation, and at an inflated price.
i would keep searching. i really enjoy kata beach, and for me, there is plenty right there at that beach. though i do self drive, so can drive around the cape point and get out. if that didn't offer enough, HH definitely will not. though you can hop a train, and hit some more southern beachs, AO Manao, ban krut, bang saphan, will have more relaxed feel, but offer even less than HH, except excellent R & R.