Looks like I've gotten things working again! Somehow my node_access table must have gotten corrupted (not sure how that happened), but I just tried rebuilding my permissions with:
Administer › Content management > Post Settings > Rebuild Permissions

And that seems to have solved it, the node_access table is now populated with the one entry that you mentioned above, with nid of '0'.

Lyle, Thanks for taking the time to help me get to the bottom of this problem!
Wes