Different yeasts do give different levels of residual sweetness, but unless you exceed the yeasts' alcohol tolerance, the difference is relatively minor. Champagne yeast will give you totally dry, while Sauternes yeast will give you off-dry. If I read your question right, you are looking for sweeter than that, so changing yeast strain is not going to get you there.
The standard way to get sweetness is as you know stabilizing with sulfides and sorbate (you need both, just one won't work) and then adding sugar or honey for meads. If you have problems with sulfides, you don't want to go that route. Bulk aging until all the yeast is dead (at least a year) before back sweetening is a possibility if you are patient enough.
Question: are you sure that you have sensitivity to sulfides? The vast majority of people who think they do, actually don't. There is an easy way to test. Eat a dried apricot. A single dried apricot will typically contain far more sulfide than a whole bottle of wine. If the apricot doesn't trigger a reaction, then you are among the majority of people who avoid commercial wines because of "sulfides" who are actually reacting to some other chemical that is present.
Which is not to say that sulfide allergies or sensitivity doesn't exist, but it is pretty rare.