It may feel bad to be the person who takes out all the rubbish, but if you absolutely need to have it taken out before it starts to go mouldy, it's probably simplest and easiest just to do it yourself.
Also - the best way to get rid of this kind of irrational fear is to be exposed to it. Every time you take out the rubbish, mouldy or otherwise, and you don't get sick and nothing bad happens, you'll be slightly less afraid of mould.