The answer is
**Fish of sorts, perhaps Catfish hinted at in the 3rd verse. It could also be seafood in general. The 1st comment to this is the answer: Red Herring as it can be smoked and cured. **
I may sometimes be rather dry
But you prefer moist and hot.
The flesh is soft, you must apply
Just the right touch, don't get caught.
Smoked or Cured Fish is rather dry, but fish cooked properly is quite tasty. Fish have flesh which is rather tender. There are some types of fish which need some organs removed to be edible, as they contain lethal amounts of toxins. Could also refer to how easily fish flesh can be ruined by heavy or clumsy handling.
I must be pierced to bring you joy
My smell is known and often craved
But you should know I'm not a toy,
And there are many I have enslaved.
Fish are usually hooked and this action pierces the fish (usually the mouth) and anyone who has caught a fish does feel joy. It could also imply that you remove skin and scales to access the flesh inside. Unlike meat which smells only when it is bad and quite sickening when rotten, the smell of fish in the market does actually whet your appetite. My only doubts lie in the ending of this verse as I try to reconcile that with the concept of Sushi and the fetishes that have grown from there. Maybe the author is making a social observation about something. Depending on when this was written there may have been a seafood fab/fashion/trend in their society, hence the enslavement. Alternatively, it could be referring to health spas that use fish for ex-foliating treatments: also a social commentary.
Now what am I? Oh no, not that.
Have I pulled you into my trap?
I'm not at all related to a cat.
And found nowhere near a lap.
This verse refers to the reader's lewd thought processes. Author is making it very clear that it is not a sexually orientated riddle, although he/she acknowledges that the reader's mind will definitely consider a sexual response. Catfish are in no way related to cats apart from the whiskers found on each. Last line is confirming that a lap dance by some sexy stripper is not what this is about but there are some fetishes out there which include fish and other sea creatures. Just saying.
Some think me rare, or maybe not.
Can you rise above the cut?
Again I say, some like it hot.
What am I? Follow your gut.
In some areas fish is a delicacy/exotic dish and very hard to come by due to distance from the sea. Could also be that fish is very expensive when compared to meat. "Rise above the cut" plays on the pun of a 'cut of meat'. It ends on a pun as well as fishing requires gut to make a catch as well as referring to your now rumbling stomach ;)