Republic isn't just about how the representatives are elected. You can have a republic with a popular vote. More importantly a republic is about the rights of the individual that can't be voted away by a majority. So I read that as the United States guarantees that each state will be a republic, as the states also cannot vote away the rights of the individual.
Democracies have laws and rights, and by definition those rigthts are at the peril of the majority. Under our founding and Republican form of government, we view our rights, as derived from God, which cannot be voted away.
Those rights, innumerated in the Federal Constitution cannot be voted away, anything else is open to Democratic rule.
I direct you to the 10th Amendment to the Federal Constitution.
I'll offer two questions
In plain words what does this say
,the right of the people to keep and bear Arms,
shall not be infringed.
What does that mean, and under what process have they been eroded?