This isn't entirely true. I think you are conflating the Primary election with the General. You will most likely have Republican and/or Democrat ballots during a Primary because you are voting for your party candidates or your party ballot issues in a primary.

In the General everyone (in the same precinct) will have the same ballot because you can vote for either party in the General. Even states with jungle primaries get their party's ballot during the primary, but not the general.