Why not try this one
if you are concerned about drippings, why not try snuff (also known as dry snuff) instead? If you go for an unscented variant, the flavor should be so subtle that the deer should be less likely to smell it, because it basically stays up your nose.
I think if you are a smoker or otherwise used to tobacco, the effects of using no tobacco will be probably more negative, because you will be experiencing some kind of withdrawal syndrome.
What you could also try is to use "Snus" a kind of chewing tobacco from Northern Europe, which is a variant of chewing or dipping tobacco, which leads to less saliva production.
I hope this helps.
P.S.: Somebody is drinking beer while hunting? Now, that sounds more weird to me than dipping... ;)