I agree its tough to get a basic little baby doll for a very young child. I do home childcare and wanted several simple ones, as mine always end up naked anyways! They like to put some clothes on them, but mostly just wrap them in little small bandana (or smaller) sized blankets I made from fleece. Bandanas are great too..we have about a dozen of them! And the little Velcro fabric diapers..my 2+ year olds love to change their "stinky" diapers??? So I watch for those at thrift stores and garage sales.
But as far as the baby dolls...I have the Fisher Price ones. I went to Kmart. I hadn't been in for about 10 years, but online they were the only local place that listed them when I was looking after Christmas. At the holidays I got one at Walmart for my great niece who just turned one and she loved it. I looked online and this local Kmart was the only place with any in stock...I went in and bought all 6 they had and my daycare kids love them.
These have soft bodies, but the head, arms and legs are that hard plastic. I got one set of the twins, and 4 single dollys and spent just under $60 on all of them. Worked for my needs!
They are the Fisher Price Little Mommy Newborn dolls