Your yeast and your recipes and everything else should have very little if anything to do with the carbonation. The Palmolive and dishwasher however are overkill and might even cause problems as they require so much rinsing. Best way to clean your bottles and equipment is with hot water and no-rinse sanitizer as per one of the following:
A) If never cleaned before or containing dried beer or soap residue, soak in hot water then give a good scrub with a bottle brush. Then sanitize with StarSan or OneStep.
B) If collecting bottles as you drink them, it's best if you always pour your beer into a glass and then IMMEDIATELY rinse the bottle in hot water 3 or 4 times while it still contains wet beer residue. Then when it's time to refill, your bottles are already like 99% cleaned and can most likely go straight into sanitizer (StarSan or OneStep) on bottling day. Same thing kind of goes for your fermenter -- once you have emptied it, IMMEDIATELY rinse with a lot of hot water before any beer dries up inside it, and it will be ready to sanitize next time. Soap might not even be needed unless you have a lot of junk clinging to the sides (e.g., dried yeast near the top).
As for the priming, if mixed well, you'll never have a problem again with bulk priming. This is the best way to go:
In a standard household glass (about 12-16 oz size), mix together about 1/2 cup water and a scant 1/3 cup cane sugar per 2.5 gallon batch, and boil in your microwave for about 2-3 minutes. Ensure the sugar is fully dissolved, then allow to cool for about 15 minutes. Then add to your fermenter in bulk right before bottling, being careful to swirl or stir it in so it doesn't all just sink to the bottom.
If you do all that, you really shouldn't have any problems. Try the bulk priming. You'll get more consistent results.