The only way the fuel leaves the bowl is leaking past the bottom plugs, evaporation, or draining back after shut-down. The drain back issue can be solved by removing the clip from the needle, but they only drain back to that level anyhow.
This new fuel evaporates pretty quickly, but still should be around after 2 days. My own carb takes about a week to evaporate the bowl dry.
There should be no rich running symptoms on cold starts if the fuel is evaporating from the bowl. It would have to be draining into the intake to do this, or the choke isn't set up quite right.....Cliff