Do you shake like a bitch during lock-out? That was my biggest issue, the OHP and pushpress are pretty good for hitting the core.
Also, if you dont shake violently, that doesnt mean your core is good to go, either. Things like ab rollouts, full rollouts, TRX/blast strap fallouts are good for hitting your core in a very similar way to the OHP.
If your seated OHP is much better than your standing I would guess your core needs work, maybe 20% or so difference. Make sure that you stay at lockout and tighten everything up for 1-2 seconds after every press, too.
Of course, the small muscles are probably an issue too. This press will move slowly for a long time, so be patient. Youre also standing, which of course is harder. Make sure to keep your lats taught and strong in general.