How do you know if your dough is too wet?

I completely agree with Smaug. Many factors affect the overall hydration of a dough. I bake in Denver (high altitude, low moisture in the air) and use directions as a starting point only. Flour seems to absorb lots more liquid than in printed recipes (perhaps the ground flour I use has lost moisture during transit to this dry area and storage here). I always need more liquid for baked products using flour. At first I thought that it was a problem, now I just think of it as a reminder to learn how each dough should look and feel. I have had some failures, but those were when I was feeling restricted by the recipe. I think it has made me a better baker.

More or less in response to the original question- I did have some times when I was experimenting with baguette doughs (which my dentist has now forbidden) when the dough got wet enough that it wouldn't hold it's shape; that, I will say, was too wet. I don't have one of these baskets, but I do use a cloth for rolling doughs and I almost never wash it- it, or the corner of a drawer I keep it in, did once get an infestation of some sort of weevils, but usually I just shake it out and it's good to go

To echo both Smaug and Lori: maybe (regarding hydration). If you have trouble handling the dough during fermentation and shaping processes, then your dough might to too hydrated (this is very subjective), and you might consider taking down the hydration to something you're comfortable working with, and then increasing from there. At about 80% hydration for a mostly white flour bread, this is not a super easy dough to handle (at least for me, it's still something I'm practicing).

But if, as Lori said, you have no problems handling the dough up until it goes in the proofing basket, then better flouring might be a simple fix! This might be gross, but I don't wash my proofing things after every use. They get a buildup of flour and they're never sticky - I shake/pat them down well, and let them dry after each use. They're a little damp but never wet. One sturdy linen couche I have even came with instructions to avoid washing. Between washes, no visible molds or mildews grows on the cloth, and they smell like cloth and flour (no funky smells).

The author does point out in the notes on the recipe that the stone ground flours she uses tend to absorb more moisture than other types- if your cloth is coming out really wet, I think it likely that you do need to make the dough a bit drier. I distrust bread recipes that call for strict amounts- a lot of factors can have slight effects on how the dough handles and particularly how it hydrates- the flour, what's in the water, the temperature, the humidity etc.; you're in a lot better shape if you can get a feel for the dough that works for you.

I don't think the trouble is that your dough is too wet, because that problem should make itself apparent during the previous resting/folding. The problem is likely due to the dry cloth and flour wicking moisture from the dough while chilling. You need to really rub the flour into the cloth itself, not just sprinkle it on the top. The idea is to work it into the fibers and create a flour coated surface to act as the barrier between the threads and dough. I also find using rice flour mixed with a little bit of regular flour makes it less likely to stick to the cloth. Then the skin on the surface of your dough should be able to resist giving up so much moisture to the liner. That's another thing to keep an eye on. In the final minutes of kneading, before the dough goes into the basket, you need to roll/pull the dough to make it have a tight cover of dough on the top- and secured by pinching that at the bottom. When you transfer this dough ball to the basket, that tight skin will go in facing down, with the pinched bottom part facing upwards. Remove it by putting your baking sheet on the top of your basket and holding that in place while you tip the basket and sheet over. Then give it a bit of a shake and your dough should come out. You don't have to be too gentle with that shake, and your dough should recover any slight loss of air while it waits for the oven to heat up. To clean the liner, I usually take mine outside and give it a good shake to start. Then I leave it soak in a large basin of warm water for a bit, and then rinse it well under running water, before hanging it to dry. You really don't need soap, just lots of warm water and time. Any bits of damp dough will dissolve and rinse away. Your liner ideally would be made of a tightly woven linen or muslin, and that holds up pretty well to washing by hand. If yours is made of a thinner material, it might be worth your time and effort to replace it. The liner doesn't have to be attached to the basket either. As long as it covers the basket inside, it can simply be laid in it and smoothed out as necessary. As far as the final hydration of your dough itself, you can adjust that as you like, but generally a wetter dough makes nicer textured bread. You don't want it any drier than it has to be to hold the shape you want to give it. Dry dough equates to dry crumbly bread. From the sounds of it, your problem isn't with the hydration of your dough. I think if you get rice flour worked into your fabric, you will solve most of that.