"What happens if a state like Minnesota deadlocks? Is there a mechanism for breaking the tie, or does that state end up effectively not casting a vote?"
They can't cast a vote. If the House deadlocks, the VP elected by the Senate could be "acting President" for up to 4 years.
Acting President McCain?
Heads up. With things this close, remember Nebraska and Maine are not winner take all. They divide up their districts and apportion EVs accordingly.