I stamp with liqua-gel food coloring. The key is not to apply it too heavily. (It looks like it's pretty heavy in your photo.) To ensure a light, yet uniform coat, I just moisten an uninked stamp pad with the coloring, and blot off any excess before inking the stamp. I find that the coloring dries quite quickly this way, regardless of the brand used. (I have used many types of liqua-gels and gels, but I most often work with Chefmaster.)
Airbrush coloring, especially if alcohol-based, will dry faster, but I don't use it because I find the coloring too loose, making the stamp more prone to sliding around while stamping. But some people seem to like this approach - most seem to airbrush it directly onto the stamp.