my problem in the quant. section and sentence correction... i am not a native speaker, but my grammer is very good and i got my bachelor from the states................
For quant I agree that Manhattan is very good. I think your problem would be Data Sufficiency since Problem Solving is mostly easy. Also remember that for problem solving OG is quite useless since most of the questions are very easy.
For Sentence Correction Manhattan SC is a very good book but I don't think it's the best book to start off with (unless you are getting 620+ in your diagnostic test). it can give you too much information the first time you read it and can actually unnerve you. Since you've got a 320 on your diag., I would suggest that you start off with a simpler book that gives you all the rules (something like the SC Grail) first and then go through Manhattan later once you develop a comfort level with SC.
320 to 700 is a steep jump, but I'm sure it's possible.
Good luck to you!