You might be able to pull this off with a lenient DM
While lifting a cover from the darkness-ensocerelled object will cost you your free object interaction, the intent of the rules is that just dropping an item is free and does not cost an object interaction. This was confirmed by Jeremy Crawford on twitter:
The intent is that letting go of something requires no appreciable effort. But picking it up does.
Now, Crawford's missives are not official Sage Advice rulings any more. But considering he is one of the game's rules authors, and his intent was for dropping to be free, you may use this as an argument to convince your DM that it should be.
You then could have your darkness emitting object, for example, on a string over your shirt, pull up the shirt over it with your free object interaction to remove the darkness around yourself, use your action to attack, and then let go of the shirt for free, which due to gravity will slip back down exposing your darkness bauble yet again.
I personally would probably not be so lenient, as I think this is to some extend an exploit, and unfair to players who invest into abilities such as the blind-figthing feat to be able to operate in magical darkness. The DM might rule that holding the shirt up while attacking is hampering you and imposes disadvantage on your attack, making this more symmetrical. On the other hand, you do not benefit from advantage during your attack as someone with blindsight does against a blinded opponent, so maybe it is not that bad, and the DM might be OK with it. It does cost you a casting of darkness, after all.