Usually fluorescent primers are purchased with the fluorescent dye conjugated to the oligo by the manufacturer.  You could order one PCR primer with 5'-FITC and the other primer with 5'-Cy3.  The manufacturer performs oligo synthesis, performs fluorescent dye conjugations according to their standard method, purifies the conjugates, does QC check and ships them to you.  After PCR, your PCR product could have a FITC-labeled strand and the other strand is labeled with Cy3.

Your method used labeled and unlabeled nucleotides.  During PCR if you incorporate too many fluorescent labeled nucleotides in a small region, you can get self-quenching by too many dyes in that region.  Also, incorporation of FITC- and Cy3-labeled nucleotides can result in FRET between FITC and Cy3 dyes, i.e. Cy3 emission is enhanced at the expense of FITC emission as long as the two dyes are within the Forster radius.