Your paint should never be so thick that it causes that effect. If it does you need to thin the paint more.

Use the 'wash' technique. Thin some grey/black paint, and then run it over the panel lines. Do a search on google.

Paint has different 'strenght'. Generally its lacquer, enamel, acrylic. If you use an enamel wash on acrylic, its gonna reactivate the acrylic paint and mess up your model. Acrylic however sucks as a wash because of the surface tension of water, so...

Go to Ace Hardware and buy a bottle of Future Floor Wax (55 ringgit I know cos I ran out and bought another bottle today). Paint your kit, let it dry, spray Future on it, let it dry, and then you can do an enamel wash without really worrying about paint reactivation. Remember to really thin your wash as well!

Oh, and yes, you can use tamiya primer. I'm so broke I use bosny automobile primer.

Kinda dangerous to use thinner, mild as it is. No left over sprues to test it on ke? I never use thinner to strip paint, I use some branded over cleaner. Works great and never ever harms the plastic.