I could have written your post a few months back! I started the same way, walking on the treadmill then I got all gutsy one day and decided to run for a minute straight--it about killed me!

Once I was able to do a full minute running/5 mins walking for the whole 45 mins, I decided to take the leap to HIIT (High Intensity Interval Training)--mind you, I still kept it at a walk/jog pace--but by the time I finished out the 8 week program I was doing 15 30 second jog/walk intervals. I'm now on Week 7 of C25K and lemme tell ya, I love this program!!! I have a week of 25 minute runs ahead of me and I'm ready for them (granted, I'm not going to win any races, but I'm getting faster

)! I never, ever thought I would be able to run for 25 mins straight--and I have a bum knee to boot! (strength training works wonders--haven't had a single problem and I don't wear a brace or anything when I run)
I think you're doing a great job! Keep it up! You'll be amazed at how much your body gets used to running and breathing will soon get better--just take it slowly and you'll get there.
Congratulations on your addiction!
Btw, let me know if you want the link to the HIIT program and I'll send it your way.