I'm with you on this. One of my studio's instructors asked me if I had problems hearing the beat. I've listened to enough music that I can now hear it in many songs (not all, yet), but I often lose it after I start dancing. He calls that "Don't bother me, I'm dancing". I'm getting better, as I can often recognize when I get off the beat, though getting back on is still a challenge.
Your situation is more basic. You need to listen to more music. Don't listen to the radio when you drive. Listen to the music you will dance to. Listen not to the tune, but the underlying cadence (not all beat comes from a drum). You should count until you are so comfortable that you can pick up the beat in the middle of the song and count it accurately. Some instructors will tell you not to count when you dance, but I'd suggest you tell them to go pound sand until you "get it" (politely). Eventually you will, and then you won't need to count (physically). Like any other skill, practice is the best medicine.