getting from Al Mazra'ah, Karak to Sheikh Hussein Bridge

Greetings all,

I plan to go to Israel via the Sheikh Hussein Bridge on Sunday and am trying to figure out the easiest (and least expensive) way to get to Jerusalem from Al Mazra'ah. Is there a way to get to the bridge by bus that avoids Amman? Al Mazra'ah is located right at the southern tip of the Dead Sea. Also, is the Sheikh Hussein Bridge similar to Allenby with a bus to shuttle one across the border? If so, is the cost the same? How much should a taxi cost from the border to Beit She'an?

Thank you for your help.