I don't know that this is the right place for this post, and I haven't read all of these threads so it may have already been stated, but here's what I think happened.
I believe that Casey drowned Caylee for whatever sick reason (to get back at Cindy perhaps or because Caylee was in the way of her new life) in the family pool. She freaked out afterwards (thinking OMG what did I just do - did I really do that [no insanity here]) and that's when she had to really get a plan going (to cover her butt - afterall all she cares about is herself). She put her body in the back yard until she could gather the items she needed to dispose of her (blanket, bags, laundry bag, duct tape, etc). Tony L has stated that he was with her on the 16th (and I can't remember if the 17th) and no Caylee; he said he skipped school, IIRC.
I believe that she decided to pin this on Jesse if it came down to her butt getting in trouble. I suppose she thought she could go on forever and no one would ask her where Caylee was and expect her to deliver her.

(It's late and I'm beyond exhausted so hopefully this is making sense.)
Anyway, I think her calling Jesse on the day he was working when she ran out of gas was to let him know where her car was at (that way she could say well he knew my car was at Amscott - and she has stated that - so he must have put Caylee there).

She also has said that he has a key to her car (does he? IDK). But she did 'conveniently' need to use his shower while Tony L was in NY over the July 4th timeframe. Could she have planted something, to include a copy of her car key, at Jesse's?
I do believe she told LA of her problems - it may have been while she was out on bail. I think that is why they bought the story of "you can't trust Jesse". I mean, why not, she trusted him just 2 weeks prior enough to use his shower? :waitasec: Thus, no reason to look for Zanny the invisi-Nanny. IMO, they know Casey did this and they are just looking for anyone else to pin this on so their 'precious' daughter doesn't have to rot in jail where she belongs.