What’s the most effective way to promote an Android app besides adding to Google Play store?
Hey, here are some more points that will help you to increase adoption rate of your app. http://www.geekpersuite.com/grow-adoption-rate-mobile-app/
Check on the below links to know how to properly promote an Android app -
Ways for promotion
Sites to choose for promotion
Optimizing the app's UI to appeal to the users at first sight
if it is shareware you can make a smaller freeware and buy now to unlock other features?
you can post in android forums like latest release section.
provide few serials for a review.
make youtube tutorial
Latest Android Apps
I disagree with offering a crippled version of your software. If you're going to let users test run your software, you better let them see the full version. Also, concerning the promotion of your app, see the link below. It's the one and only guide you'll need.
Remember that word of mouth is the best promotion (or possibly the worst if you do it wrong) that you can get. Go to the community that your product is targeted for and sponsor it yourself. For example, the signature to your post on most forums.
Android Developer Center - Promoting Your Apps