Dennis,
Various seed companies offer cover crop mixes that work well for cool season planting. I've got about 10 lbs that I'll be planting tomorrow and Saturday. I have no affiliation with this company, but this is where I order my seeds, as well as the microbial inoculant needed to help the legumes fix nitrogen. Here is a link to the cover crop mix I build from:
https://www.groworganic.com/soil-builder-mix-raw-lb.html
I buy about 10 to 20 lbs of it at a time, and then I add a bunch of my own seeds to it as well. I'll throw weird stuff into the mix like Tansy, tillage radish, or a bag of black-eyed peas from the grocery store. It's the inoculant that I need from them, as my soil doesn't naturally have the right bacteria needed for nitrogen fixation. Hopefully, in another couple years, I won't need to order the inoculant -- it will be in my soil in sufficient quantities.
No, you don't need to mow it all down --- at least not initially. You can cut down small areas and then plant your seeds or transplant your plants. Then, as the plants continue to grow, you just hack back the cover crops that are crowding your garden crops. For plants like watermelon or other vining veggies, I use
newspaper to cover the cover-crops as they slowly work their way out into the cover crops. That was you don't disturb the soil, you keep all that N
underground where it needs to be, you bury that biomass under newspaper (Back to Eden style), and it's not much work at all. For tall, leggy plants like corn or okra, I'll make a space in the cover crop for the initial row, but as the okra grows, it'll outcompete the cover crops. Eventually, I'll just stomp them down into place between the garden rows.
Vetch (purple or hairy) is a climber. It's the one exception. I tend to yank it out of the ground, or it will tangle up everything as it grows up the corn like a
trellis.
Best of luck.