This problem has existed since I was sending 5" reels to stations and they carted them.
So this is coming out of a 3rd party studio, into commercial radio stations. If it sounds fine coming out of your studio, but distorted on air, the fault lies between the two. It could have something to do with; a) the operator loading the automation machine, b) levels the operator uses, c) levels the jock plays back on air, d) how the limiters and compressors in the transmitter chain are set, e) all of the above.
The correct answer is E.
Clean recording, with a reasonably high bit rate, means you are already doing everything you should to ensure a smooth playback sound. The cement heads who take it from there are screwing it up.
How would I suggest they clean it up? Stop using a compressor across the master out in Production if they are doind so. Have the jocks pay attention to levels while on air (yeah right ...). Don't set the transmitter chain so that you have the loudest station in town, rather set it so it's the best sounding station in town. Less limiting, less compression at every link in the chain.
Chances of that happening? Well unless you can convince the PD and Engineer that you are going to walk away with 70% of their revenue if they don't clean up their signal chain, resulting in the GM firing them ... I'd say the chances would be exceedingly slim.