many people want to be reassured that they are dealing with a big company.
They key in such a relationship is to know why they want to deal with a big company. And in order to give an accurate answer, one needs to know what do you sell, to whom and where, things that you don't specify. In my answer below, I assume that whatever you are going to say it's true. Not fabricated, half true or assumptions. I write it here, to avoid writing it in every other sentence. Also the answer is based on my experience in software industry.
I know my finances are sound, but how can I reassure the client that I am not going to disappear tomorrow?
As others already said, you can't. Because it's psychological. You can't convince someone who doesn't trust cruise control to use it. They trust the accelerator pedal that's been around for over a century. Cruise control? What's that new devilry? Of course it heavily depends on how you define disappear. Like physically? Or win the lottery and go on long vacation to the Caribbean? Or just be bored and say "that's not for me any more"?
In any case don't spend energy in convincing them to prefer you instead of a big company. One thing you can do is to show them the other side of the coin. Find a couple of real examples of companies (preferably in your industry) that ceased operations and left the customers hanging. Not dragons and fairy tales, just the facts. Suddenly a one-man company is not as bad as it sounds. You can also try to explain that this isn't a side gig, your existence relies on what you do. At the very end you can always ask them. Something like "What does a big company offer you that I don't?". You have to be prepared for their answer. Even if you have to say "sorry I can't help you with that", you now know your weak spot and you can think of a way of fixing it.
Bonus part, if you work in the software industry and you sell the product, not the service, you can always argue that software doesn't rot. They can still use it after you disappear. They won't have upgrades, but they aren't empty handed either. If they pay you upon delivery (and by delivery I mean acceptance) it's a win-win.
How can I word contracts so they don't appear to come from single person, yet aren't deceptive?
That's fairly easy. Form a company and get all your contracts through the company. Usually contracts have clauses like
"Your next big talent Ltd" from now on the "Provider" and the "Cash Cow Holdings" from now on the "Customer" agree[...]. There is no number of people mentioned. I know as I've done this. Just to make myself crystal clear, customer knows your are on your own, the contract is drafted in a way that doesn't show that. Another thing to have in mind is local legislation. Where I come from, a one person limited company is different than a more than one person limited company and that appears in the company's full name. In other countries it's not the same.
Also you get one more point in the previous question, as sometimes the existence of a company itself shows that the person behind it already has invested time and money to create a brand.
Is it ok for me to "hide" that I am, ultimately, small time, even though I pour my heart and soul into this?
Don't do that. Instead show them that while on your own, you've done A, B, C (your portfolio) which are amazing and that is because you pour your heart and soul in what you do to quote you. Focus on making your brand appear to have a team behind. When they come to you (or you go to them) and they ask you "so where is your team" you answer "you are looking at it". Some will say "meh". Some will say "that's interesting" or even better "You did all these by yourself??? Get outa here!". You direct your efforts towards the second and third type of customers.
 If you have many customers, hire a contracts lawyer to draft you a template. It might be pricey but it will payoff in the future. And you will get another point on your first question.