There is also the option of 3/4 length sleeves. They were the traditional length sleeve for nurses (before the age of scrubs,) to deal with just the problem you are describing. Fob watches pinned to the uniform were also traditional to avoid having the watch interfere with scrubbing up. These concerns were not considered obsessive, either. Also, a ring of any complexity (as opposed to a simple wedding band) tends to be a habitat for germs.

A bar soap that used to be popular with nurses was Palmolive. (Less drying.) Whether or not a soap is "antibacterial" is really less important than how well the hands are lathered and rinsed with whatever soap. Another traditional - and good - idea is to avoid touching those spigots with bare hands.