My 2 1/2 year old is clingy too. She constantly wants to be by my side, etc. While I am in the kitchen, I have her sit at her little table and give her a sheet of stickers. You can get the huge books of hundreds really cheap -- like at the dollar store. It is good for her small motor skills to pick the stickers off the sheet and she LOVES putting stickers on her pictures. She'll usually do this for a good 15 minutes or more so I can get something done. The dollar store also has bags of the foamy stickers. She LOVES those too. She chooses who will get the picture she's made and she gets excited to mail it to grandma, etc.
Also, she is starting to not take naps anymore (or she won't go to bed til really late!!) so I've decided to have her spend an hour of "quiet time" in her room. I tell her she has a choice to take a nap or play quietly in her room. Of course, she doesn't want to sleep, so she reads books, plays with her baby dolls, play food, etc. She normally won't even play in her room. This gives me time to sit and watch my shows, take care of the baby without feeling guilty, work on the computer, etc... I put a timer in her room and tell her when the timer beeps, she can come out of her room -- that way she doesn't come out of her room to constantly ask when she can come out. We JUST started doing this and it is working really well so far...
Also, one other thing is just to stop and hug her. They really need that sometimes. I get carried away doing other things around the house and she can get insecure sometimes. When I don't take time to hug her, she just gets more insecure and becomes more clingy. I just have to remind myself that she is just a tiny little human and she has needs too. My job as a mommy is to meet her needs too.
Hope that helps! Good luck. I know it's really frustrating...