If you only bring along a shovel for avys, ditch it unless you're traveling with other people that carry all of the typical avy gear (transceiver/probe mainly). In fact, you may want to ditch it altogether because of the wooden handle. However, if you bring it for snow caves, etc., then fine.
Ice axes can certainly get lighter (7-10oz). Unless you're doing ice climbing, or doing serious hard snow alpine climbs all the time, you don't need a burly axe. Heck, Camp even has a steel pick on one of their 8oz axes, but for what you say you do – likely not needed unless you run into ice. Keep in mind, though, that a lighter an axe takes more effort swing it, since it will likely require more swings to get good penetration/steps cut.