Hello, Jaideep!
First off, nice job on the 680 (even if it's not where you're trying to end up). The 48 in Quant is particularly strong, as you likely know, and indeed, to get to a 720, you'll need Verbal to improve by a few points even if you do manage to get a stronger Quant score.
In looking at your ESR, I saw a lot of the same things that you already noticed, but let's break them down/contextualize them a little more:
CR is absolutely an issue in Verbal. High level first: if you're not pressed for time in the Verbal section as a whole, you should definitely spend time at least attempting the longer CR questions. It may well be that you still need to sacrifice one such question to make time for the others, but essentially if you're taking 30 extra seconds on each of the longer CR questions, then you can sacrifice 1 of those questions quickly and buy yourself enough time to do 4 others. This is especially important because as you might have seen in the ESR, the questions you were getting wrong in the second half of Verbal were actually a bit easier than the ones you were getting right--that's not good, as those questions can tug your score downward particularly hard. In terms of what's meant by Analysis/Critique: yes, a lot of strengthen/weaken/assumption questions will be in that category; however, to get really clear practice with that topic, use the Official Guide's online learning platform by using the code at the front of the book. In that platform, you should be able to select which questions you want to practice according to the same categories you see on your ESR (this also applies to Quant). So you should be able to ask it to just give you Analysis/Critique CR questions if you like and thus get a clearer understanding of what they are.
RC is something you really need to go well to get a high score in Verbal. And while it may seem weird to have missed a bunch of Identify Stated Idea questions, it's not super uncommon at all. There are a few reasons why this can happen, but the two most likely reasons are that 1) the harder versions of these questions involve a lot of inference and 2) the answer choices can be particularly tricky in these questions. I'd watch out for that second one especially as you prepare for your next exam--whenever you get a question wrong in RC, make sure you've fully understood not only why the answer you picked was wrong and why the right answer was correct, but also why you were tempted by the wrong answer or put off by the correct answer. Basically, take every question you get wrong while practicing RC seriously and review it carefully.
In SC, high-level questions tend to test meaning a lot, so you can practice those. However, as mentioned before, the online platform for the Official Guide should allow you to filter for Communication-type questions.
Finally, Quant: Geometry accounted for 3 questions on your exam, so indeed, it wasn't a major topic and shouldn't be a primary worry. You missed 2 out of 7 questions in Rates/Ratios/Percents and 2 out of 5 questions in Equalities/Inequalities/Algebra, so those could be good topics to practice. Again, the online platform can help with this. That said, I think that the biggest thing you should be working on is hard questions. The questions you missed were pretty high-level questions, and it's possible that you missed them less because of the topics tested and more because of their actual difficulty. Luckily, the online software will allow you to filter for Hard questions only, so make some time to practice that stuff.
Overall, I think the jump you're trying to make is very doable. You should focus more heavily on Verbal (especially CR/RC) than Quant (probably in at least a 2 to 1 ratio). Also, if you took a course with us, you should be eligible for a free PEA (Post-Exam Assessment), even if the course was a while ago. Reach out to your instructor or to
gmat@manhattanprep.com to set one up.