Skilfully cover table-top with oil in a dry environment (8)

I believe the answer is:
adroitly
'skilfully' is the definition.
(I've seen this before)
'cover table-top with oil in a dry environment' is the wordplay.
I cannot quite see how this works, but
'a' is found in the answer.
'table-top' could be 't' (top of the word 'table-') and 't' is located in the answer.
'cover' could be 'oil' (oiling is a kind of covering) and 'oil' is located in the remaining letters.
The remaining letters 'dry' is a valid word which might be clued in a way I don't see.
This explanation may well be incorrect...
Can you help me to learn more?
(Other definitions for adroitly that I've seen before include "With skill - dilatory (anag)" , "Dexterously" , "With skill and subtlety" , "Quickly and skilfully" , "With deftness" .)
