Practicing with a metronome can enable you to express the grove/feels better.
If your internal timing is good and trained, it will be easier for you to add groove without the thing falling apart.
You didn't really specify though if your problems are related to physically nailing a feel or groove or to interpretation of the song at a whole - these go hand in hand imho.
If I play a cover song, I try to find out as much about the song, the singer, the intention, I try to grasp the meaning and lyrics of the song as much as possible.
From this information I try to come up with a mental image, a feeling, a memory that I can connect to the song. I try to imagine myself in the singer's/protagonist's place, and what might have compelled them to write the song.
On a more technical side, I try to use all this to guide my groove/feel.
Often I use pictures of movement to help me grasp a certain feel better. Could be a train, surfing, running, a river flowing, the ocean waves, a mild summer rain... - whatever comes to mind.
This will all move your focus away from the hands and the instrument, more towards the actual music and what you need to put into it.
Take a very simple bassline, and try to play it with diffrent emotions - play it bitter, play it sweet, play it desperate, play it light, play it in every way you feel or can imagine.
Another thing that helps me (with covers) is to watch the actual musicians and how they move, or in some cases to watch how people dance to a certain type of music. I move a bit like them and it instantly sounds more like it.
