After much trial and error on using Ansible modules with Digital Ocean, I thought I would post an update so that others don't go down the same path and get as frustrated as I am.
As some instructions point out, you should use dopy 0.3.5. However, both 'private_networking' and 'unique_name' do not work. I didn't thoroughly investigate, but it appears that a change in the API led to the breakage. I have found that if you use dopy 0.3.6, 'private_networking' was fixed, although I'm not sure if anything else was lost.
If you are creating a brand new domain record, this works. If the domain already exists and you use 'state=present', you will get an error. Looking at Github, it appears that a fix was submitted back in January, yet it sits unreviewed, going nowhere.
I'm pretty frustrated. I don't know Python, but considering learning it now. And trying to think of some other solution to using Ansible with Digital Ocean. :(