My best tip is to see the doctor. Last year for a few months when I was in between insurance coverage- i got a cold....and I thought I could treat it on my own and be fine (I also have asthma) but I wound up getting severe bronchitis and a plethra of other issues because I was stupid and didn't see a doctor when I should have.
Once I did get to the doctor though it wound up being a lot cheaper than I thought it would and she gave me tons of free samples and some good prescriptions so I got better. I did wind up loosing about 2 weeks of work though while I recovered (could have avoided all that had i gone in the first place
So definitely get to the doctor before it gets worse if you think it merits a trip- especially if you have asthma!
What inhaler do you use? My doctor recently switched me off of Albuterol and onto Xopenex http://www.xopenex.com/
(It works a lot better for me and doesn't raise my heart rate through the roof like others tend to do.)
When I went to see her while I was sick she told me to get some Mucinex DM (That it's just as good as most of the prescriptions out there,but a lot less expensive.) She also gave me prescriptions for H-C Tussive cough syrup and Tessalon Pearls (the generic). They both worked very well and didn't have any problems with my asthma. (If it comes down to going to the doctor, maybe you could ask if those might be a good match for you).
Also- if you have GERD or any sort of reflux- make sure that stays under control between medicine and diet- GERD left untreated can make your cough a lot worse.
I found a nice cup of decaf green tea with honey and lemon always helped my throat!
Dan Active is great too as far as immune systems go- maybe you could give it a try! It's really helped me
The Vicks Vapor Inhalers are AMAZING too - those always get me through the night when I have a stuffy nose and can't breathe!!! http://www.vicks.com/vapor-inhaler-info.php
(If you have a walgreens drug store near you -their store brand is just as good!)
One thing I would say- I've never had any luck with Zicam....it doesn't work for me so i wouldn't recommend it.