What to say other than “Sorry”?

Instead of constantly apologizing, let your actions do the apologizing for you.

Here's a list: Haha pardon if the words are kind of awkward!(1) This one's the best - Take your mistake as a joke! Take it lightly. Just laugh it off and she'll find it funny too. Plus, happy people are charming.(2) "Oh oops!" - For if you drop something or bump into her(3) Explain clearly and honestly why you did what you did and directly express concern though use of language, etc. Then say you meant no harm by it. But if you did something wrong, say sorry. That's what sorry's for!(4) "Oh. Derp." Sorry to introduce gamer language here, but it's effective and another form of taking life lightly! Besides, is that TF2 I see? :P If so, Medic! Yus! Best class! Good for you! If you don't play TF2 and have no idea what I'm talking about... Just... Just forget about it lolBut honestly I think that laughing it off is by far the most powerful substitute. Laughing never can really be overused, and it makes you seem more confident, relaxed, and happy and makes you feel that way as well. Just look at it in a positive light and see that it's funny when you've made a small foolish mistake that's not hurting anyone! But when you really are sorry, explain your side of the story and explain that you didn't mean to hurt her at all. For girls, apologies are big things, and usually 'Sorry.' for being late for a date or something. Chances are, she isn't even bothered about your repeated 'sorry's, but rather, your lack of confidence! Just learn to ease up and relax a little! Life's funny! Tiny mistakes can be funny! Don't make yourself miserable and beat yourself up over it!

It all depends on the situation but try talking about why and how you feel. But somtimes sorry is about all you can say but try like sorry i didnt know such and such hurt you ill remember that or sorry i didnt mean it like that i ment.... explain but dont defend your self and try to come up with a game plan to not let it happen again

Ask her. Ask her, "Ok, i hear you, but, what do i say instead?".

Some of my favorite things to say when I do something clumsy/dorky:

"Aw, crap."
"Boo hiss!"
"Aw, sh*t!" (only with a long southern drawl like Ellis from L4D2 so it comes out AWWW SHEEEEET).
"Fudge monkeys/muffins!"
"Tartar sauce/fish paste!"
"Ruh-roh Raggy!"
"Ah, f**k."
"WILSONNNNN!" - that one especially if I drop something while cooking. Was the best when I had dropped an egg on the floor when making pancakes one morning. Dunno whether it was actually that funny or if my bf hadn't had enough coffee in him yet, but he laughed hysterically for about 15 mins.

When things aren't going well or get awkward:

"We'll be alright if we can just get back on the expressway." (yeah, I quote Blues Brothers. Especially comical when we get turned around while driving)
"I'm not a complete idiot. There's a few pieces missing."
"Yeah, that could have gone better."
"My bad!"
"God d*mn it, Caboose!" (Yes, I also quote Red vs. Blue!)
"Yeah? Well, you smell funny!" (during an argument).
"Parley!" (Also in an argument on occasion)

And just some random stuff, like when I'm about to make us late because I'm looking for something, I'll yell "Why is the rum gone?!" - I quote a lot of movies for random moments in life that they kind of apply to; sometimes yeah, sometimes no, but it usually earns a laugh and breaks tension for me and for those involved.