I am having a difficult time getting first aid and skinning to move in combat.

I am currently hunting Cougars.

I use Preserve then skin the mob. For some reason neither skinning or first aid ever go above Dabbling.

Am I just not killing them fast enough or am I doing something wrong?

I'd have to see your ranks to see if something was out of whack there, but as a catchall suggestion for skinning, make sure you arrange fully before you skin every time. And make sure you keep the skins to scrape them later, normally the only time I actually lock skinning is when I scrape after hunting. It's also good for free mech ranks too.

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Your mind hears Sahfra thinking, "Always the temples. We will burn your temple to the ground."
Your mind hears Zaslamel thinking, "shutup you undead wench"
Your mind hears Malfrien thinking, "hiya Zasl."
Your mind hears Zaslamel thinking, "Hiya Malf"

OP, try getting a small bleeder on your back or abdomen and use harvest. FA will just stay locked all the time and you learn more thanatology. Once than locks, arrange and skin.