Alice take heart.
Nobody has cleared bindweed in half a year and it sounds like you have a lot.
In my first year here I used a trowel frequently to keep removing it. Two years of that, is worth it in the end, crops grow meanwhile.
I would pull up all the surface roots where you plan to crop now.
With more cardboard & compost if you can, but on less than half of your allotment!!! keep poly over the rest.
Less is more, crop intensively on a smaller area.
