I disagree that buoyancy and swimming are unrelated. Logic says if you are negatively buoyant some of your swimming energy has to go toward keeping you afloat. Positively buoyant people put all their energy into forward motion. Just my theory--I was briefly on the HS swim team (1970, and since then didn't really swim at all until 2009) but recently got a sense that if I slowed down I'd be sinking. Anyway, taking swim lessons is the best advice. You are certainly by no means the exception. Having observed the OW 200 meter/yard swim test for 4 years now, I would say the 7 out of 10 students do not swim with a "proper" stroke, though it is rare that anyone "fails" (ie., they "gut out" the 200 meters). It is my opinion that one who can't swim with a proper stroke is crazy to take a scuba course. BUT, the statistic I mentioned says I may well be wrong. If you don't take lessons, maybe these tips from my competitive swimming brothers may help and save you time & money:
First, it seems you would be able to pass the 10 minute float test with the method you describe. Or, you can do a search on "drown proofing" (big breath, head goes under, kick arms & legs fast after a bit to surface and get another breath, etc.).
SWIMMING:
1. Don't kick too much/too fast. You get 75% of swimming power from your arms (unlike scuba, where you use your legs only-- except in odd situations. As mentioned in another post, don't "bicycle" kick--legs straight as possible, kick straight up & down, getting power from your thighs-- same whether swimming or scuba). You may only kick once or twice at most per arm cycle. Otherwise, too much/fast kicking knocks yourself out and doesn't increase speed much. Keep in mind the 200 meter swim is not timed-take it easy with proper technique. My mom was one of most proper swimmers I ever saw, but with her speed, she would never win a race.
2. Arms: When returning each arm to the front keep them as close to the water as possible. You get nothing when they're fanning air. When arms in the water pushing back, there's most of your power. Dig deep in the water and keep them close to your body, even touching your butt if you like.
3. Head/breathing: Don't lift head out to the left to breath and next time to the right, alternating. Excess energy. Chose one side to lift head to breathe and as with arms, lift your head out only as far as needed to catch a breath. Any further head height is again, more wasted energy. You may on occasion choose to skip a breath--keep your head face down in water for 2 or more arm cycles. Or not.
4. Turns: I can't do the fancy swimmer's under water flip thing, but when you get to the pool end, push off real hard for the next lap. Use the glide you create as a little "breather" between strokes.
All these tips took me from not being able to finish the divemaster 400 meter timed test to doing it in about 12-13 minutes, a decent time score. Of course, once you get your technique down you have to do a lot of fairly regular pool lap swimming to get your SWIMMING muscles up to par. Any other exercising will help a little to get into general good shape, but does little to improve your swimming.
I got most of these tips from my 72 year old brother who still swims competitively (2 or more ocean miles) and wins medals. Of course, he's done laps almost daily for over 40 years.
Personally I don't like that students can either do the 200 swim or 300 with mask/fins/snorkel. They are not related, with the 300 IMHO being way easier and doable by a "non-swimmer" who can kick. Also, it may or may not be a choice depending on the instructor--another interesting idea. But that's neither hear nor there.