Are these signatures legally binding?
At law, if you sign a document, you assert that you understand it and agree to be bound by it - it doesn't matter if you have read it.
Now, everyone knows that most of us sign many documents without informing ourselves fully of the consequences. Judges both know this and, no doubt, do this themselves. However, this is not an excuse for a practical and pragmatic reason - if we could disclaim our signature just because we claimed later that "I didn't read it", then signatures would become worthless as evidence.
While you have to have the opportunity of reviewing the document (and I'm sure they'll show it to you if you ask), you don't have to review it. But, if you don't, that's on you.
If I was shown one of these documents under oath, I could honestly say I had no knowledge of what the content was. Can I have 'had the intention' [sic] to sign a document without ever seeing it?
I don't understand how you claim that you lacked the intention to sign something that you signed. Are you claiming that your writing hand became possessed by a ghost or a demon and signed the document itself? Does your hand sign documents while you are asleep?
You consciously picked up a pen and signed the document - therefore, you intended to do so. Provided that you were not coerced, drunk, drug-affected or under some other legal incapacity, your intention was clear. And no, the fact that it takes time to read and understand the document and we are all time poor is not coercion.
I only signed a blank screen with limited or no other context. Should I be refusing to sign?
I can't give you legal advice.
You need to do your cost-benefit analysis. Are the costs of taking the time to read the document, either thoroughly or by skimming, more or less than the risks of agreeing to an unknown liability?
If you are buying a house, a car or a mortgage, almost certainly yes. These are big, long-term commitments that will substantially impact your finances for many years.
If you are signing up for a pay-by-month phone or internet plan, maybe not. The commitment is small and easy to get out of, the terms between providers will be very similar and there is a fair amount of consumer protection and the risks are correspondingly low.
If you are signing a medical consent form? Well, in most of the world this would fall into the mobile phone plan category. In the United States, it's probably more of the mortgage category because involuntary servitude appears to be legal there for prisoners and people who owe money for medical bills [that's a joke - just not the funny type of joke].
Even if they show me something, how do I know that's what my signature is being applied to?
Because they told you it was. Legally, that's called a representation and one of the grounds for voiding a contract is if one of the parties made a material misrepresentation about it.
Edit to address comments
A number of commenters have raised issues about proving that the document purported to be signed was the document that was actually signed, either in general or in the particular circumstances. Those are irrelevant to the question!
On the facts as stated: I signed a document I didn't read, am I bound by it? The answer is an unequivocal and emphatic: Yes.
The facts may be disputed, and the method used may impose some difficulty on proving the facts if they are disputed, but that is the question: If I can prove I didn't sign the document, am I bound by it? That also has an unequivocal and emphatic answer: No. And the answer is the same even if you did read it. But it's a totally different question.