Is this an appropriate drain design for adding a shower?
Except as noted here, it all appears correct if I'm understanding your drawing correctly. The shower does need its own AAV, otherwise the resulting trap arm appears to be too long. Thus you should not eliminate either vent.

Changing the toilet venting will help immensely, provided it is done right. The 3x3x2 wye must be installed rolled up 45 degrees so that toilet waste cannot enter the vent line. Do not reduce the 2" line to 1.5", carry 2" up to a larger AAV.