Heres a pic

Check on "Rigging an Eastport Pram" on this forum from about 2020, as well as some other lug rigged boats. First off, you'll need a downhaul, snugged tight and maybe even a line through the fork and around the mast. There'll be lots of pictures, but best idea is to keep it as simple as possible.    

Yes, some sort of line across the ends of the fork would help prevent the boom from kiting away from the mast.  You want something you can quickly detach when dousing sail; might be a good application for some sort of toggle arrangement.

Also, it looks like the boom or jaws might be close to bearing on the tip of the halyard cleat.  If there's any sort of contact there as the boom moves in and out, that might be what's encouraging the jaws to jump off the mast.  If that's the case (a little hard to tell from the photo), I'd suggest lowering the halyard cleat a good bit so that there is plenty of space below the spot where the jaws lie with the sail hoisted.  This will give your more room to haul the jaws down with the downhaul so as to get good tension on the luff, essential for any type lug sail.  Otherwise, it'll act more like a bedsheet.  <;-)

If you are using the simple arrangement for the sheet of attaching one end to the middle of the boom and simply holding the fall in your hand (as seems common in the example in the Eastport photo galleries), you might want to think about adding thumb cleats (not the kind which act as jam cleats) to each gunnel so you can catch a loop of the sheet underneath when you ease out for reaching or running.  This will help hold the boom down to better prevent the top of the sail from twisting off too much when the boom is eased out.  A standing lug like this needs a boom vang or some other means of controlling sail twist when the boom is eased out.  You also want to make sure that the boom doesn't rise up sharply when gybing.  Had that happen on a catboat with a 270 sq. ft. sail on a boom as long as the boat with no boom vang.  Not fun!

.....MIchael