As many 7Sagers will tell you, LG is the most difficult section without guidance and also the easiest section to improve on. It just takes practice and time. I think you can still improve significantly with 7Sage alone, without a tutor. Good luck!
I've read that it is best to take the more modern tests just before test day, as the questions and difficulty is more similar (e.g., PTs 80-92). If you haven't already taken those, it might be a good idea to use them now!
Your question is hard to answer because different sections require different amounts of time to improve. So there isn't one score that you should have by the end of June (last week; whoops!) if you want to get 165+. For example, if you had a 155 but…
I'm in a similar situation. My diagnostic was okay with a bombed LG section. Average around -2 for both RC and LR with minimal studying. But averaging -5 to -10 on LGs.
However, there's hope! Recently, I got -3 and -4 on LGs. I think the foolproof …