Continue as you have been. If you are comfortable w/him doing some jobs/chores around the home, especially those that need to be done, then go for it. However, if you don't feel comfortable w/him being there, then tell him you would prefer he not do them.
They do have a way of "skirting the truth w/o actually lying. It's sad because it's far better to tell the truth because none of us can keep the lies straight if there are too many to deal with.
Do what you think is best for YOU.
Sit quietly, the answers will reveal themselves when you least expect them to. The past is gone, the present is a gift and you need to focus on today, allow the future to reveal itself when it is ready.