Photo? Without seeing it, when poured it may not have had enough slope or pitch for run off. Instead of ripping up expensive concrete, I would install French drains on either side of the driveway to direct the runoff into the street and into storm drains.
Dusty where I live too. I just clean them out a couple times a year. Easy to do and you can hire someone to do for you if you can't do it yourself. It will help. Besides it would take an awful lot of dust to clog a 4" or 5" drain pipe. Here is how.
Naomie is right we really need a picture to give you a good answer.
Hi! Redirecting the water is indeed your best option. Check where it is coming from. Some redirection of gutters may help.
Check out the "Hydro Blox" system. If you rent a trencher, you can install it yourself. Longer lasting than French drain, and capable of handling more water.