Specific primer design
If your primers are not specific enough, you should design other primers. Try shifting a few bases upstream or downstream. Also, you can add a few extra bases to your primer, you can safely reach 68C as annealing temperature.

If you want your PCR to be able to distinguish a single base mismatch, then the mismatch should occur at the 3' of your primers, ideally one of the three last bases.

If your goal is just to quantify the initial DNA, then RT-PCR will be ok. But if you want to detect the composition of the different polymorphisms, then you should go for sequencing.