"sulk around" is not very common or familiar. I would just say "sulk".
"mope around" is OK.
The second version (with continuous tenses) seems less likely than the first. (This is because of the overall sentence form; it has nothing to do with the verbs "mope" and "sulk" per se.)
"mope" and "sulk" are not synonymous. For example, see these dictionary definitions:
Neither moping nor sulking is the most obvious reaction to being teased, though neither is impossible.
The most obvious (childish) reaction to not being given what one wants (your original sentence) is sulking, per the definition that I quoted earlier. However, moping is also possible.
In my opinion, the definition you link to for "sulk" is not great. Sulking does not really invo