Condit had obsessive secrecy - his dating rules demanding that women go out with him carrying no ID and no money is very bizarre.
I think the Levy-Condit relationship was supposed to be secret. Remember, he denied it to the Levys and to the police for a couple of months after her disappearance.
Speculating about human nature, right before the police came over to search his apt., he dumped the watchcase that had a serial number that could be traced through McKay back to him. It is behavior that looks like he was trying to erase the connection between himself and Chandra. Why bother to do this, unless he knew Chandra had been murdered and the watch was missing, maybe somewhere in the bushes in the Park, pulled off in a scuffle? I think it likely he took both the watch and the gift-bracelet he gave Chandra, both of which are missing, because he didn't want anything traced back to him. This is in line with his general obsession with secrecy. Just my opinions.