I just ran a test in my own game and had no difficulty moving Vivian Lewis from the Spencer-Kim-Lewis household in Willow Creek to Hallow Slough, a vacant lot in the same world. She used her phone to Split From Household and Move. I transferred 10,000 simoleons from the household funds, which was more than enough to pay the 5,500 for her new lot and left her with 4,500 to start building and furnishing a house. I then exited to the main menu without saving.
After reloading the same save, I used the same method to move Dennis Kim to Daisy Hovel, a vacant house that's also in Willow Creek. Again there was no difficulty at all except that I transferred 20,000 simoleons to cover the extra cost of a fully furnished starter home. The house cost 15,891 and left him with 4,109 as expected.
I didn't bother trying to Split and Move to an apartment because PipMenace already did that with Liberty Lee.
I don't know why you're having difficulty but I found it quite easy to Split From Household and Move to either an empty lot or an empty house. My best guess is that you just aren't giving your Sims enough money to pay for the lots you want them to move into, as already mentioned by jpkarlsen .
