Here's another Kentuckian who has never heard ish cream, or any other s regularly changed like that, and I grew up listening not only to KY accents but that of my Tennessee relatives. This use of Southern kind of makes me think of (erstwhile) Vixy's problem with the use of the word classical: there are MANY "Southern" accents, and possibly how one says the s sound may have to do with something more than just being from the South. For example, some people say worsh for wash, and I suspect that has more to do with the educational level the speaker comes from.
I will concede that--maybe--I've heard s softening towards almost a z, as in eyez cream. But no h. Definitely no sh sound.