If you are married or in a committed relationship, then you are NO longer in the dating game and you should not pretend to be. Sending mixed signals to work buddies and new friends of the opposite sex can be fatal. Secretive behavior will only add fuel to the fire if your partner demands explanations.
In a relationship it's important to establish boundaries and know what is acceptable. Talk about your concerns if you are uncomfortable before they escalate into something bigger. Keep in mind that this is a two way street; and if you are seriously flirting too and suspicions arise, get ready for confrontation because it is coming. If your gut tells you he's totally trustworthy, you might want to give yourself a 'self-esteem' check - jealousy on your part could signal a low self image rather than a cheating partner. BUT, if he is acting shady even after you confront him, his behavior may mean he is not really committed to you and he's keeping his options open. If that's the case, you know what to do. . .