She almost did, his body found by accident by road repair people 7 wks later... HOW did that happen??
How did she get outa state without a tail?
What convinced LE that she was not worth following, that she had left him locally?
So many questions and so few answers to this aspect.
BBM
This next part is totally my own opinions and thoughts based on the question.
LE assumed she had dumped GS locally. LE probably never considered that LS would do something as crazy as go get GS body AGAIN and take him all the way to Florida(of all places) because:
1. The new category of GS case/status (missing endangered, not just a runaway), and they could tell LS pretty much knew she was suspect #1.
2. All the media attention on the case.
3. Past instances of anyone doing that are so rare that it just wouldn't make sense to jump to that train of thought.
4. LE assumed that being able to tell where she had BEEN, thru gps locations, would lead them to GS either alive or (most likely) dead.
Honestly, off the top of my head I can only think of one case where a killer transported a dead body any considerable distance after the murder. Sometime last year a guy killed his whole family then drove to a police station several counties away with two of his dead relatives in the car. But that's really not the same because he wasn't trying to conceal anything. Iirc, he had one in the trunk but the other was right there in the front passenger seat and was obviously the victim of a brutal, bloody attack even if not obviously dead. Any cop along the route could have taken one look and pulled him over. (I'll cite a link or something to identify this incident when I find it.)
I said all this to say that the cops heard hoof-beats and thought "horses" because they're in Colorado and zebras just aren't the first animal they'd think of... Unfortunately, LS has such an alien mentality they probably should have thought of something you find in an episode of Star Trek.