I've nearly completed recovering my Tower trainer but I'm having trouble getting the monokote to stick to the edges of the engine bay. I'm not sure if old fuel is keeping the covering from sticking or I'm doing something wrong here. Any suggestions?

I am assuming its not fuel laying on top, as you have recently stripped the plane..to recover. So I imagine its IN the wood? There are different ways to pull fuel oil out of wood.

I used to use a stain removal aerosol called K2, you sprayed it on and let it dry out, then dust it off.

Most types of body powders will pull moisture , heck I don't know what cat litter could hurt !

I am afraid I can't recall all the other variations to remove fuel oil..If you search it you will be able to find other ways.

Also they make some brush on adhesives to help covering stick, sig stixit and a red baron has a covering adhesive..it might be worth trying.

Your name makes it pretty obvious you served in the military..thank you so much for doing the hard job of protecting our country.

Yep foodstick nailed it, K2 is a spot remover it works. Any good solvent such as acetone will remove oil...and CA glue ...and the glue to monokote.
I have used CA glue to fasten edges in such places too.

An old trick that may help a great deal is to coat the area to be covered with any of the alaphatic glues (like Titebond II) with a very thin coat after you have removed all the oil/dirt that you can. Let it dry then iron on your covering in the normal manner.

I'll add to whats written above that in general its hard to cover wood that has epoxy or some sealer glue on the other side.
I'm using more heat on the sealing iron or heat gun on places that are epoxy covered like fiberglassed wing joint, I dont get it right the first time its trial and error process, also the trim solvent can help too.

A real cheap oil removal solution would be to mix rubbing alcohol and corn starch into a thin paste and apply it to the area. Apply the mix and let it dry and brush it off. Continue the process until the mixture dries into a white powder, at that point nearly all the oil should be removed out of the wood.

sand sealer over the oil soaked wood. I have a tower trainer that had a leaky fuel tank and it flooded the entire bottom of the plane. Stripped the covering. Let it dry out. Sanded out the warps and puttied it up. Several coats of sand sealer. Recover with monokote. Just watch the heat because the sand sealer can off gas causing bubbles. You can try titebond as well but when I tried it out, it actually melted/bubbled off from the heat of a covering iron.