M.B.
L.,
You. Are. NOT. A. Bad. Mommy! I think part of the clingy-ness is the age. My daughter is 2 1/2 and is just as clingy; and she's the little one!
Here are some things that help us with her:
1) Tell the 2 year old that you're going X-Y-Z and will be right back
2) Tell them you're going to wherever and will be back at lunch time (or some other time reference he'll understand)
3) When I leave I go to a certain window and we wave bye-bye and blow kisses through the window.
4) Just leave even though she's screaming, it's a ploy to see if Mommy will come back/not leave (my son was a master at this one in daycare)
The last one is the hardest, but with my son by the time I was out the door, around the wall and into the observation room he was done crying and was playing with toys. He's now 6, in kindergarten, and can't get on the school bus fast enough in the morning.
Good luck! I know how you're feeling with the cling-on clinging all day every day.