Jordan Valley / Sheikh Hussein border crossing to Jordan

There are also the Nazerene tours busses which go from Nazereth to Amman. Last time I took one, the other way (in May) they split at the border with busses to Sakhnin and Um Al'Fahm. I'm not sure if their times will fit in with you, but you could get a sherut to Nazereth on the Saturday (they definately run) and then the direct bus on the Sunday. I don't have the number to hand but if you phone Nazerene tours, or ask someone else to, they'll give you the info. They don't have a website. They are mostly run for students from the north who study in Jordan, which is why they're not so well known. I'm also sorry that I don't have the sherut number but I know that they run too (or take a bus after dark).

G

Yes, there are sheruts from Tel Aviv to Nazareth on Saturdays. As on every day of the week, they go when they are full, and I have no idea how long they might take to fill up.
From Nazareth to Beit Shean your only option on a Saturday is a taxi.

Search the forum for Nazarene buses; they have been mentioned here before.

Another option would be to go via the King Hussein (Allenby) Bridge if you already have a visa for Jordan. The sherut from Tel Aviv to Jerusalem runs on Saturdays. You would need to get from the sherut drop-off point in the Jerusalem city center to the Damascus Gate (about a 20-minute walk or a taxi ride) and take a bus or service taxi from there to the bridge. Visas for Jordan are NOT issued at this crossing, and there are often long delays there.