Re: 5k Race Help - I'm breathing too hard!
I admire your willingness to come here and ask basic questions--and I admire the conviviality of coolrunners; if you'd asked the same question at letsrun.com, you'd be full of thrown knives, and leaking, right now.
The answer to your query is both quite simple and fairly complex. You've set a goal of 20:00, and goal-setting is indeed an essential part of racing well. But as far as I can tell, that goal--as a short-term goal rather than a two-year goal--is quite out of line with your current abilities.
Here's the simple answer: your breathing rate is way out of line because you're running the first part of the race at a much faster pace than you can possibly average for the totality of the race. Your intemperate early pacing is spiking your heart rate, not just your breathing rate. In fact, it sounds as though you're maxing out fairly quickly: running the first mile of your race like you'd run a hard 800 interval, achieving maximum heart rate, then slowing until the pain (and breathing) is barely manageable and running the remainder of the race in that pained condition. That's bad racing.
Skilled racers--and I consider myself a skilled racer, although not a particularly fast one--can race 5Ks around 95% of max HR. (I'm a consistent sub-20 5Ker, by the way: not by much, but still.) But the difference between 95% and 100% is huge. And it's important to feel the difference, and avoid 100% until, at most, the final 800 or so. The worst possible way to run a race is to max out early, then try to "hold it." You're dragging a high anaerobic load with you the whole way. That slows you down AND hurts more than starting easier and finishing harder.
So far I'm just speaking to how you're actually running the race. But that's where you should start: by forcing yourself, in your next 5K, to run the first mile at the AVERAGE pace you were able to sustain last time around. This will actually be a little slower than ideal, but it will transform your race experience. You'll actually warm into the race, feel what it's like to work towards a hard-but-sustainable pace, even in a race of that relatively short length.
Bit by bit, you'll figure out how slow/fast to go out in that first mile in order to run hard-but-sub-max through the 2 mile point. The differences are subtle, but real. When you learn what 95%--or 92%--feels like, you'll learn how, for example, to flow up hills in mid-race, rather than attacking them and paying the price at the crest. (Later on, of course, you can think about race tactics; first, though, you need to get yourself under control.)
Then, of course, there's the question of training. For good 5K racing you need an aerobic base, some tempo work (sustained 20-30 minutes at a pace that works you hard but DOESN'T let those telltale flames burst through your chest), and some 2:00 - 3:00 intervals at race pace. A 22:00 5K is about 7:00 pace. But you may actually be a bit slower than that. I think you'd greatly benefit yourself for the next little while by running intervals of various lengths around 7:30 pace and getting really used to what it feels like to work at that pace.
As for aerobic work: well, your 60 min runs @9:00 pace are a good start. Combine them with 20-30 minute tempo runs at, say, 8:00 or 8:15 pace. And be patient! You'll achieve your goal, by and by.