How do I get my duplicants to finish a dirty job before showering
You can accomplish this by setting up a different schedule with a section of "work" before "bathtime" for particular duplicants that have a high tidying priority, see the following examples:
As you can see, Travaldo is a duplicant that has the highest tidying priority but has "work" before "bathtime". Couple this with the priority system and Travaldo will always mop liquids before anything else. Make sure they have enough time to clean up messes before bathtime, otherwise the same issue you described will sprung.