She's reacting to these changes, and the unknowns are scary. Things to reassure her of (not during pick-up, but anytime) that helped my kids feel more secure:
1. That you will always be there for her (you'll never leave her, cover that abandonment fear) and even though you're S, SHE is taken care of ALWAYS 2. NONE of this S was her fault 3. BOTH parents will always love her
THIS WAS BIG FOR ME: They look to us closer than we realize. If she senses tension or sadness from the parent she's leaving that night (not at all suggesting you had any), she will likely be a bit clingier. If both parents are upbeat and positive about the exchange, it helps A LOT in the security department. (Not perfect, but definitely helps)
Also, keeping "transitions" (or hand-offs) quick, and don't linger and feed into the thoughts of "missing" you, which will cue the clinginess. It will get better over time. The difficulty is the guilt....more on the parent side. Kids are usually over it when they get in the car. Sad, but true.
. 