Do you use cover crops or mulch in a cold climate?

Question:
How do you keep your soil covered?

Details:
If you have been gardening for a while, or if you have been researching soil health and natural ways of gardening, you will find that all of them tell you to keep the soil covered.

My question is how best to do this in a cold climate. I understand the benefits of cover cropping, and after listening to a podcast recently, I think it is more beneficial than mulch alone... but I don't feel like I have time in my climate. Generally from the time the frost threat is over until the frost threat returns... I am growing food plants of some kind. I'm not sure there are any cover crops that would grow in that shoulder season?

What do you do?

Weeds grow as a cover crop. Depending on the weed, I leave the root intact and just chop and drop the top part of the plant. Then I plant my vegetables. After harvest, it's not too long before snow falls and that covers the soil. I'm also using spoiled hay and wood chips as mulch. I don't feel like I've really dialed in a system.

I'm having the same problem. There is no time after harvesting to sow something to germinate and cover the soil over winter. Even though my winters are very mild (intermittent light frosts only), they are cool and so wet and windy that most plants just sit and rot.
At the moment I've been covering the beds with seaweed and bracken mulch after harvest. That way I feel I am adding more nutrients and carbon for soil structure than any plant will contribute that will grow over winter. As I am still at a remedial stage for my soil I'm pretty sure that this is the best option for me currently. Since the root crops tend to get left in the soil until I want them (another advantage of a mild winter) the mulch sometimes is quite late being added, but the crop itself acts as a cover crop I suppose.
I am considering undersowing my crops before harvesting with a cover crop to overwinter. Apparently this can be a way of getting the next crop established before the first matures, but I suspect it isn't going to be as straightforwards as 'the books' say!

I am in zone 5b, a little warmer than you at 5a, so this may not work for you. I mostly use winter rye, winter peas, and hairy vetch in beds that are targeted for warm season crops, like tomatoes, peppers, or squash. I plant the winter cover crop in October, the earlier the better in Oct. It doesn't do too much until late March and April, then it takes off like gang busters. I aim to terminate it by mid May. Some years, by mid-May the rye is not in the milk stage yet (that's the growth stage where crimping or cutting it will not regrow.) In those times I cut it, and then cut up the roots. I aim to plant my tomatoes/peppers by 3rd week in May. I use the cover crop I cut for mulch.

I also have some dedicated beds just for cover crops. I plant winter rye in those and then harvest the rye for mulch the following May. I then turn right around and plant a multi-species cover crop, with legumes added to the mix, to regenerate the soil before the rye get's planted again in the fall.

If you are growing crops until frost kill, you have the main benefit of cover crops—living roots in the ground as long as possible. Mulch can compensate for the ground cover, but the roots feed the biology.

These days, our winters (at lat 52.48, so too little light for much to actively grow within a month of the solstice) vary between cool and wet, and some frost (occasionally to -10 C).

I also use a mixture of volunteer plants, green manure crops and organic mulches to keep the soil covered.

Because our winters tend to be relatively mild, we have some persistent overwintering deep rooting volunteer plants such as perennial grasses, dandelions, docken, bindweed, perennial thistles, creeping buttercup, hawkweed which I try to remove from cropping areas each growing season (the creeping grasses & bindweed are hardest to remove completely!)

I tolerate ragwort as I'm not making forage hay, so we get cinnabar moths.

Volunteer ground cover plants (with more or less spreading habit) which I love include various speedwells, our geranium: herb Robert,  scarlet pimpernel, garden & woodland forget-me-not, chickweed and fat hen (the last two being good greens).

Plus we have a wonderful spreading woodland-type strawberry which came to me via my late Mum - her neighbour had weeded out a tiny seedling, which we thought looked like a true strawberry, and saved. It has stacks of tasty small fruit (pea to kidney bean size) from June, with a few coming as late as September! This is a good groundcover, and seems to actively deter creeping grasses, so that's great.  It partners with the less strangling type of pink (field) bindweed we have
I tolerate field bindweed more: it is a good nectar flower, and spreads much more slowly than the larger white (hedge) bindweed.
 
Green manures I sow .. mostly phacelia / blue tansy from home saved seeds, which is half-hardy so only sometimes lives through our winters. Sow mid-Spring to late-Summer for us.

Mulch is biomass from the above plus Russian comfrey and clean sheets of cardboard where I'm trying to knock back perennial grasses and bindweed in growing beds, or it's too late in Autumn to establish living rooted cover.

For 5 months of the year, snow covers my garden. That seems good enough for me.

As soon as the snow melts, the multi-species self-replicating cover crop starts growing prolifically. (also known as weeds.)