Hi !!...Yes you have done right thing of stopping usage of soother..
This is his primary dentition ...so don't worry ..this habit of soother won't affect his permanent teeth. No need to correct overbite at this stage .
Correction is done only in permanent teeth.
Next Steps
keep a check on usage of soother or other habit like thumb sucking .Mouth breathing can also cause proclined teeth
Health Tips
don't worry about genes ..it's not in your control.