Starting GMAT Prep with First Practice Test at 680

Parts of the GMAT and GRE are quite different, and some test takers are better at one test than the other. It sounds like the GMAT is the much better choice for you. A 680 is a great starting point, and there isn't much value in spending more than 2 hours per day on study for most people (there's a limit to how much we can absorb in a day without leaving our brains some restful time to process what we've learned, and for most people, 2 hours is about as much study as is helpful), so you should be fine, though if you could add two weeks to your timetable, that might make a difference.

You're right near the top of the scale in Quant (Q51 is the max), and while you could realistically hope to improve your Q score to a Q50 with good prep, improving your Verbal score will make the bigger difference to your overall score out of 800. Naturally if you can improve both, that would be the best thing to do, but I'd suggest you make Verbal a higher priority. As far as how to prepare:

- for Quant, I'd suggest going through each topic one at a time, first learning concepts and theory, then doing a lot of practice questions. For practice questions, official questions are the most realistic questions you'll find. The explanations in official books are not great, however. If you have trouble with any official questions, you might look up those questions on this forum, and if an Expert you know and trust has posted a solution, you could refer to that.

- you'll probably also want material that explains Quant concepts. At your level, whatever material you use you'll be able to get through quickly. No matter what material you choose, be sure it's material that teaches math, and not material that focuses on "strategy". At the high level of the GMAT, they're testing your conceptual understanding, and some of the prep company "strategies" like backsolving are of no value. I've written a set of books covering all of GMAT math, specifically for high-level test takers, so feel free to contact me if you might be interested in buying those. If you use something else, just be sure it's material useful to a test taker at your level (and you might want to first confirm the harder questions from the source sufficiently resemble the harder questions you've seen on your official test and in official materials -- some companies just design inordinately complicated questions and call them 'hard', which is never what the real GMAT does, and using unrealistic questions is an inefficient way to prepare).

- for Verbal, you're doing well, but will surely improve with practice. I'm a Quant specialist, so you might follow the guidance of a Verbal expert here more closely than mine, but the one thing I'd strongly advise is that you focus almost exclusively on official questions, which are much better than any prep company questions you'll find. You should be able to make a lot of progress purely from self-study if you're good at analyzing your mistakes. You'll likely be able to pinpoint patterns to your wrong answers in, say, Sentence Correction - particular constructions or grammatical errors you're not attentive to - and address those issues, so that you don't make similar mistakes in the future. My feeling is that SC is the area where people make the most improvement through study, and RC is the area where people make the least, so I'd study accordingly, assuming you're roughly equal at all three types right now (if you're notably weaker at one, focus more on that). You may find you need a resource that explains SC or CR concepts, but I'd not be the best person to ask for advice about that, since I don't keep up with the various Verbal products on the market.

Good luck!