I would probably not transfer in your situation.
Many of the more grade-deflating top 20 schools have a tough adjustment period coming from high school, and then GPAs subsequently jump up, which is what I experienced and generally observed in my peer group. 66% of your GPA is still undetermined (or 75% if you take a gap year) and the hard part is behind you, so I would expect an improvement if you learn from your experiences and adjust accordingly. Also, you're not even in that bad of a place right now. 3.65 cGPA/3.4 sGPA isn't ideal, but according to the 2017-2017 AAMC chart the matriculant average was 3.7 cGPA/3.65 sGPA so you're not far off. And that doesn't even take into account any leeway given to GPAs from top grade-deflating schools due to their reputations, although how much of a bump that gives you is always debated.
It would be one thing if you didn't particularly enjoy your current institution; then you could go ahead and look only at GPA-relevant factors. And yes, going to a state school will probably reduce the difficulty in getting good grades, and being less expensive is a nice feature. But you say that you do really like the place where you're at right now.
You only have one life to live, and there's more to it than being pre-med. In all likelihood, both roads will probably lead you to an MD acceptance, but diverge when it comes to tier of school or time it takes to get there (number of gap years). However, it is really difficult to predict what exactly will happen in each situation, i.e. going to state school could result in a higher-tier acceptance because of a higher GPA, but other "soft" factors at your current t20 institution could actually do a better job of making you attractive at top schools. Since the potential outcomes aren't really clear, I would go with what you already know: that you love the place where you are at now. I don't think that's worth sacrificing for estimated increases in competitiveness, but it's up to you.