It would be a very confusing situation for the children to have their father living in their home while their mother entertains a new boyfriend in the same home. The dating guidelines when you have children say to not involve the new person in your children’s lives until there is a level of commitment by both persons in the relationship that points toward marriage. Otherwise, there is the potential for children to grieve the loss of another father-figure/mother-figure if the relationship breaks up. If this occurs frequently, they can begin to distrust the commitment of a parent in their lives.
Resolving the boundary issue of pursuing a divorce yet living in the same home is also an important issue to resolve, as this too is not only confusing to the kids but to the adults as well.