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I can't seem to raise my LR score - currently about -5 per section per PT. Not worried about LG or RC much.
Normally I go back and write out why which one is wrong and why the correct answer is correct. It seems to be a variety of things that trick me each time, so I'm not sure how else to move forward.

Would -2/-3 be too unreasonable to attain before the June exam? I have finals coming up so I have to balance prepping for both, and I'm not sure how to go about improving LR in such a short time span.

It's possible to improve if you study enough. You should be reviewing your wrong answers as much, if not more as reviewing new questions. But with finals I doubt you'll have enough time. Unless you are scoring at or above your goal for a few tests the last week of the LSAT, I would push the test until October.