There are plenty of things you can do that don't come within a hundred miles of hazing. My university did not have sorority houses either. We had suites (which I know are more personal and private than classrooms) but it was never a challenge to come up with bonding activities.
Consider doing a social service project together. We used to visit an area nursing home/retirement center during its "social hours"...omgosh, we had a blast. (The little old men could be the biggest flirts and would always try to get our phone numbers, but they weren't serious -- it was all in fun.) We also got together and made Valentine cards for patients in the children's ward at an area hospital. One year, we combined forces with a fraternity and went Trick-or-Treating for Unicef. See what organizations in your local community need, or offer in the way of group fundraising ideas or ways to donate time. I was Social Service Chairman one year, and believe me, to "do unto others" is a great way to unite your chapter -- you're doing something highly positive and good feelings about each other and your organization are really enhanced.
Are you allowed to bring refreshments into the classroom where you hold meetings? As initiation approaches, have a study session/pizza party where you quiz pledges on the history of the chapter and other things they have to know. (Of course, make sure the door is tightly closed. ) We used to do that in our suite, and it was always fun and relaxed, helpful to the pledges and always good for laughs when we'd wrap up with crazy things like trying to recite the Greek alphabet backwards.
You want to encourage togetherness and bonding. You know, scavenger hunts actually put pledges in the position of competing against each other, so I'd avoid activities of that sort. Why not have an optional Sisters Night Out every few weeks for both actives and pledges? Catch a movie, grab a bite to eat, attend a sporting or cultural event on campus. This encourages friendship, is fun and puts your sorority "out there" for others to see, which can be positive for future recruitment.
Well, these are just a few ideas. I am sure you will get many others. Have fun!