Can we dig a pond instead of draining excess water from our backyard?

That might work actually, especially if you install a pump to circulate the water. I wouldn't consider it a DIY project though--I would at the very least talk to a couple of landscape engineers to get an idea of what such a project would entail. Also, talk to your county officials about building codes, etc.--you don't want to go through all that trouble to find you've done something illegal. At the very least, there might be permits that need to be obtained.