Ok. I know this sounds really, really gross, but I've been in the same boat as you and was desperate to try and conceal it better...
basically the cold sore was half on the right lower corner of my lip, and half on the skin below it. It was already at the hard and crusty stage, so on top of Abreva-ing it to death, I had to wear a lot of chapstick. I noticed this made the hard, dead skin flakes of the sore soft. Since it was the flakes themselves that made it nearly impossible to use concealer on it, I removed them...with tweezers.
You have to be very careful and make sure you're only removing the dead skin that is ready to come off, and remember to do it gently. If you have to tug it off, it's attached to healthy skin and it will HURT. I've done that other times and have ripped off a super thin layer of healthy skin on my lip and let me tell you it's the 2nd worst stinging pain besides a paper cut IMO. You don't want to make it worse! Having a cold sore is bad enough lol!
Anyhow, after I did that it was way easier to try and cover now that it wasn't all flaky. In my case I wanted to wear lipstick, so I but a light layer of lip balm on, followed with a light layer of lip primer/neutralizer (I used Benefit Lip Plump). Then, I used concealer to cover the part of the sore that was on my skin below my lip. Lined my lips and applied lipstick, and it wasn't noticeable at all.
If you don't plan on wearing lipstick, try and play up another feature to distract from it and use concealer where you can (ideal if your coldsore is somewhat on the skin, like mine)
and if you get them often, I agree 100% with Tara_Hearts. I take L-Lysine and it has helped reduce the number of sores I get.