> Previously I would just click "New Tor Circuit for this Site" a couple of times, and I would hit on a country relay that would allow me to reach the site that was blocked in another country.

What happens when you click on "New identity"?

Can you provide detail suggesting that the problem is not with what exit router your circuits are using (which could certainly lead to blockage, e.g. I often encounter blockage from UK sites when the exit router is not registered in the UK) but with the entry guard?