@OP
I don't think there is a problem here.
Me
2S + H
2O
2 Me
2S=O + H
2O
Me
2S=O + H
2O
2 Me
2SO
2 + H
2O
How to avoid hydrogen peroxide in product? Stop the addition. Dimethyl sulfide is a pretty good reductant. I don't think you need to do anything else.
@discodermolide
I'm not seeing this. What is the mechanism here? You can make the anion of hydrogen peroxide without decomposition. If you want to do a conjugate addition to an enone, use hydrogen peroxide and (catalytic amount) base.
I can write a balanced equation for the formation of oxygen and water without base. Therefore, a catalyst only need to change the reaction path, not the stoichiometry. As described, this is a stoichiometric reaction, "If you stop the addition of LiOH the reaction stops." I do not understand what is being oxidized.
If you do a Payne oxidation, if the reaction is basic, then oxygen is released.
RCN + H
2O
2 (base)
RCONH
2 + O (bracketed)
2O
O
2If not basic, the peroxy imidate can be intercepted to form an epoxide.