By now, you and your partner are probably both gingerly touching each other with your tongues. If you are both comfortable, feel free to explore each other's mouths with your tongues. Keep in mind that it's always better to do a little bit at a time to get a feel for it. Remember to occasionally pull your tongue into your own mouth so you do not bombard your partner. Small, exploring French kisses are what you want for the moment.
Swallow your saliva. French kisses are bound to get a little messy, but control it as much as you can by swallowing your saliva. If you can't do this while kissing, do it when you are giving your partner a short break from the French kiss. This should be a short, momentary break and you should continue to be engaged with your hands.
Continue exploring each other's mouths. Keep a soft tongue. For most people, it is a major turn off to feel a ridged tongue in their mouth. It isn't welcoming or sensual in any way. Remember that your tongue is one of the most sensitive parts on your body and should be kissed as such. Think of licking ice cream from a soft-serve cone. You don't want a lot of pressure. Just enough to lick a small layer of ice cream off. That is the same thing to think about when kissing.