weed control

Hi, hoping to get some advice from any missoula area readers out there. We live in the bitterroot on the east side. We have very poor alkaline soil and my husband in effort to establish a garden had soil from a nearby defunct dairy hauled in a few years ago. This soil has produced fairly well, although it is also fairly alkaline and I have been trying to amend it. The worst problem though is the horrific weeds that came with the soil. I gave up last year and resorted to landscape sheeting but even that is difficult, weeds come up between the rows faster than anything can sprout. So I am open to any questions but am wondering specifically about two options. First one - what about newspaper or cardboard with a layer of straw around and between the plantings. I am concerned about using straw sprayed with herbicides, does anyone have any local sources for safe straw? The other option I wonder about is a heavy cover crop of something like clover. Where would I find seed locally and when would I plant considering the local climate? Any answers or suggestions would be appreciated, otherwise I am afraid the weeds will defeat me this year. And by the way the garden is fairly large, about 65 X 65 feet and includes a 60 X 20 hoop house, so solutions need to be fairly large scale. Thank you!

Do you know which weeds you are having the most trouble with?

I'm sorry I don't, other than masses of field bindweed but there are lots of others. many of them are unfamiliar to me, we don't have them on our property so that's why I'm relatively sure they came with the soil.