I have a hard time believing that your top reed becomes unstuck just after putting it in your check for 5 minutes. Without that top reed completely unstuck your not going to attain the best sound the call can produce. I highly discourage keeping the call in your cheek for any extended period of time. This causes your reeds to become saturated with saliva, thus the latex looses its tone.
No turkey song on the horizon at this time....but you never know what the future can hold. As for a mouth tickling your tongue, it's probably because you have it too far forward in your mouth. This is directly connected to your high gag reflex. There's really not much you can do for that, just the way mother nature designed you.
I would try a tube call though, great tone and very versatile call.
Best Woodhaven calls? Box-Real Hen, Pot-Carbon or Vision Crystal, mouth call-Scott Ellis signature(close second, is the yellow venom, red reactor and Doug Crabtree Signature)