What antibiotics would I be able to get if I am diagnosed with a grumbling appendix, as my sister had it and it went to appendicitis?

See the surgeon. If you have appendicitis, best antibiotic is removal of infected appendix, see your surgeon.
Appendicitis. I am not familar with that term but it sounds like a mild case of appendicitis. Taking antibiotics for something like this are ill advised - it will supress the infection but not fix the problem and will simply delay having your appendix removed. If you are having right lower quadrant pain-it is best to be checked out - and if needed-have your appendix removed before it has a chance to rupture.