I know my finances are sound, but how can I reassure the client that I am not going to disappear tomorrow
You can't, because there is no guarantee. You could get hit by a bus tomorrow and you and the client need to be prepared for the possibility.
How can I word contracts so it doesn't look like a single person but isn't being deceptive
You shouldn't. The client should know that it's a one-man show and if they know they shouldn't care if that is shown on the contract—on the contrary, the contract should reflect your situation, otherwise you might get yourself into real trouble.
As an example: big companies are usually not dependent on one employee and they can carry out the contract even if one of the key employees can't continue with the project, but you should negotiate a force majeure clause that lets you void the contract with no or minimal penalties if you get hit by a bus and break both your arms.
Is it ok for me to "hide" that I am, ultimately small time even though I pour my heart and soul into this?
Personally I think that trying to give the impression that you are more than one person is not ok, especially if you are selling your personal skills and commitment. Companies are accustomed to hiring individual talent. It's better not to give false impression if you can't live up to it if need be ("You broke both your arms? Can you just pass the project to your coworkers? No? But you said you were 100% sure your company could meet the deadline!")
It's somewhat different if you sell something like an online service through a web site where you might not want to emphasize the size of the content provider.