Lee should avoid any romantic or otherwise “close” relationship with Carly.
And in a nutshell here is why:
Carly already knows about Lee’s criminal past and can expose him at any time and on any whim. So she already has him by the balls. If they decided to get intimate she will then have him by the balls and the heart. She at anytime can make any choice she wants and will have to become number 1 priority to Lee trumping even Clementine. To make matters worse if they have a falling out then in her emotional rage or vengeance she may expose Lee’s past.
I hope the game gives Lee the option to become "close" with Carly, the romance drama would be ironically comforting in an otherwise hopeless time.
Lee should avoid any romantic relationship with Carly at all costs and keep her at a arms distance.