Nadenken is 'thinks' meaning ponders, reflects, I believe. I remember it by seeing it as being without any particular object - although a Dutch speaker will probably tell me I'm wrong!
I think you're exactly right. Judging from my closely related mother tongue German ((nachdenken, über etw. nachdenken)) and from what I've seen, it's intransitive and takes an object only via "over":
Ik denk over iets na = I consider sth.
Ik denk erover na = I ponder/... it.