I have to tell you, when I started out with the mindset to run a 5k, I registered for a race and went out to start running - and thought, what did I do?! I couldn't run a mile for the first couple months without feeling like I was dying! Seriously.
I started using the Jeff Galloway run-walk-run method and have had a lot more success. This week's 4 mile run was the first run that I've finished thinking "wow, that was pretty easy" and I've been running for almost a year now! So don't give up, you will get there! Just hang in there! I can honestly say that I feel your pain! I've been there!!!!
Good luck, and keep up the great work. It doesn't get easier, but you do get better! :)