I started at 220, on a comfort bike. Had the same problems- either legs gave out or lungs gave out- legs when using to large a gear and low cadence, or lungs when using too small a gear and high cadence. Did two things- got a heart rate monitor, and put "power grips" on the pedals.

I found that if I used too low a gear (spinning 80-95 cadence), my HR would skyrocket and I'd "blow up"- have to stop and recover. If I was using too high a gear (50-60 cadence) , my leg muscles weren't developed enough yet to sustain it for the climb. I found a happy medium- I try to maintain about 70-75 cadence, remain seated as much as possible and slide back on the seat. Stand every so often to use different muscles. This keeps my HR in the 80 - 85% range. Most of the hills I've done are not super long, or super steep- we have a 1/2 mile 20 % that still kicks my butt (have to stop 1/2 way up and recover for 3-4 minutes), but all the others I can now handle. The 20 % is on my list for next summer.

The BIG difference was the power grips (toe clips will do the same thing, I like the grips cause there's nothing to tighten/loosen). Lets you use the full pedal circle instead of just the downstroke. It'll take some practice to get used to pulling up, pushing forward and pulling backwards while pedalling, but OH what a difference it'll make. The other thing to do is climb hills. The more you do the more you can do.
It took me my first season to figure out I couldn't spin 80-85 up hills yet.

I used my comfort bike to ride an MS150 this year (about 3500 feet of climbs on each day), and just did a 110+ mile century on it. Trained for the 150 by riding hills every chance I got, for two months prior. I'm down to 180, and feel better now than I did at 30 (I'm 49). I'm not a strong slimber by any means, but I'm working on it. Hang in there, climbs hills, and figure out the cadence/combo that works for you to get up them, then you can start fine tuning to get up them faster/easier.