My baby won't let me put her down

Shes almost 6 months & it's pretty much been this way since day 1. Everytime I put her down she starts screaming like someone is killing her. I always make sure all of her needs are met; changed, fed, plenty of attention & cuddles, bathed, not too hot or too cold, etc. She just WILL NOT LET ME PUT HER DOWN. I don't completely agree with the cry it out method. I'll let her cry for a few minutes at a time so I can get things done, eat or use the bathroom, after I know there is nothing else I can do for her that I haven't already tried. Pediatrician said try Tylenol & teething toys, I've even tried frozen breast milk for her to chew on. But I don't honestly think that's the problem. She won't sleep unless she's attached to my boob & I have to hold her every minute of the day. As long as I'm holding her & moving, she's content. But as soon as I put her in her playpen, walker, bouncer or hand her to my husband.. she starts screaming. I really don't wanna leave her to cry but I'm exhausted & at my wit's end. I don't know what to do. I just feel like I need a minute to myself but even when someone else is holding her or playing with her, she looks at me & cries & if she can't see me, you'd think she was getting chopped up in little pieces. Lol. The pediatrician & people at the wic office look at me like I'm a terrible mother for voicing these concerns with them... Being almost in tears just trying to talk about how utterly exhausted I am. My husband said his oldest daughter (my step daughter who's 7 now) was nothing like this. He could actually put her in her playpen or on a blanket on the floor & she wouldn't cry. I know I can't be the only one. She's a happy baby, constantly smiling & laughing.. she's just so clingy & can't stand being out of my arms.