Elk could in areas with capped OTC tags. I have never heard of the Panhandle selling out though. Deer will not, they do not have a cap on deer tags
Correction, there is not a cap by unit like elk tags, but in total there is. I don't think you will have an issue though. Even when they were discounted you could still get them with ease.
Q. How many tags are there?
A. Supplies are limited to the available nonresident tag quota [LINK], which is 12,815 for elk and 15,500 for deer.