In Diablo 3, while in the cathedral, what can I do with a full inventory?

Solution 1:

There is a town portal in Diablo 3 that is unlocked when you find the Leoric's crown (I believe that is where you are headed)

You will have to continue and and chose what you will pick up until you unlock the town portal.

You could alternatively use any waypoint you encounter to travel back to town before that.

Solution 2:

There is still a concept of a town portal; you just haven't unlocked it yet.

Mundane items only sell for 2-4GP each, so you're probably safe to drop a few of those on the ground or ignore new ones you find until you reach the next waypoint. Waypoints are fairly common so you're not likely to have to wait long.

Solution 3:

You can come back for the items within a short amount of time.

How long does dropped loot last?

This question details the time it will stay on the ground while you go do other things elsewhere. Just don't change Acts or leave the game within that amount of time.