Or maybe try tracking them in a different room.

Different mic or different compressor or reverb or something. If it's standing out, maybe you should try changing the chain on the doubled parts a bit. That's what the chorus effect is and is pretty much the reason to double track the vocals. Some slight comb filtering is to be expected. Though I've doubled vocals just about every time I've recorded them and never had a problem with comb filtering. You might try a different mic for the doubles.