It sounds like developing your cardiovascular fitness could help.
I'd suggest a brisk daily walk of 20 minutes or so and maybe put a small dumbell in a rucksack to wear for your walk (maybe also put some foam in there to protect your back) to challenge your heart. Increase by 5 - 10 minutes at regular increments when you feel the current stage is no longer a challenge.
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Good advice. Especially about the weight in a backpack.
Andee have you ever been checked for Asthma? If not you might want to just in case before you start trying to improve your fitness.
Somebody else mentioned this. I have never been checked for it, I think I should get checked soon.
Andee, have you considered swimming? If you're going for brisk walks, look for a hilly course and MLG's suggestion of putting a weight in a backpack is a good one.
When you say you're out of breath, are you able to easily catch it again or do you feel like you're really struggling?
Swimming would be good, but that does mean joining a gym and wearing a swimsuit, so unlikely to happen. The suggestion of hilly terrain is an excellent one, because that's where I notice I get out of breath--going up hills and stairs.
I do catch my breath easily, but sometimes I actually have to sit down because I feel a bit sick. It doesn't sound good, does it? Especially for someone who has never smoked!