Sean, oh I know what you are talking about!!! My veggie gardens were just about the same. We didn't harvest much of anything, except beans. The horrible spring that went from winter to too-hot-summer-to-too-cold-summer, yep, yep, that be our year too. It was a nasty. We only got about 2 meals off our asparagus patch, which normally we eat maybe at least 20. That was a teller of the tale for surely.
The black plastic that you want to put down won't do a ding dang thing. Black plastic is used for solarization (well, my opinion). I don't believe for a minute that our climates get hot enough early enough in the season to do any good, it is time for planting long before the seeds are cooked beneath the soil. It really is a waster of time to do this. AND...forget herbicide crap, you don't want that in your vegetable garden, which the produce you will be consuming. I do not agree with herbicide use in the veggie garden. Some say that for example, Roundup, becomes inert, once it hits the ground. But I still have my reservations about that and wouldn't even think once about application to the food garden.
Do as Brian said, rake, turn under, over and over and over. I wouldn't bother to cover though, my preference. Like beekeeping, as a question and you will get 10 answers. There will always be weeds in the gardens, I can't imagine anyone having a weed free garden.
I have this horrible weed that has now spread to my main vegetable garden. Over the years in my main vegetable garden I have been so anal about this being a weed free garden, it almost had become.....until this year, when I was so neglectful with working the gardens.
It is common purslane aka -- Portulaca Oleracera. It is a nasty, nasty, nasty, weed. It is used in our big city for a garnish for salad, and is a highly prized garnish -- but they must grow it in controlled places. This is a flat growing plant, succulent leaves, creates a massive mat. Has absolutely invasive roots and spreads like the frickin' dickens with its millions of seeds that burst and fly everywhere. And it has. It has now invaded my main veggie garden, and this upcoming year I am going to have to work so hard to erradicate it from there, I don't know if I can, but I can be pretty aggressive when it comes to weed removal, it is just a matter of getting on my hands and knees just a little more often, if the weather is better this upcoming year, then I will have that gumption. Eeeks, I am ramblin'....
Good luck Sean, a baby in tow can really set back the time one has to spend on other things. You have had a very busy summer, that is clear and plain to see. Next year this little one will be walking around, wanting to eat all the weeds that you kick out of your garden, and then there will be even more work, hee, hee. Enjoy that baby, such a lovely little darlin'. And have the most wonderful day, great health wishes to us all. Cindi