When duck tape is useful, tape over the overflow holes and the sink drain with the condensate drain only. Check for odor at the other sink.
Condensation drain lines, A coils and drip pans inside the air handler can also get real nasty with mold. If that is from the AC unit, see if there is another trap at the air handler? It is possible that air is passing the trap next to the air handler and entering through the sink drain. Should there be a backup of the sink drain line, waste water could backup into the air handler.