just wanted to throw this out.
I have used Duro tires, many different styles and sizes on many bikes and have never had an issue with any tire I have bought from them. Presently my two 28" wheel bicycles roll on Duro Cordoba 700x38c (28x1.5") and they are stellar.
With that in mind, I have also used Kenda tires, to a similar level of success, and have not been dissapointed in them, the only issue I have with them, is getting the to seat properly. On a Kenda, I always spray the sidewalls with silicone spray, prior to inflation, so the beads seat in the correct position, a few bounces around the rim while airing, helps also.
The biggest difference I have noticed is the Duro's typically roll faster on pavement, I have no science to prove this, but the bicycle just feels like it rides with less road resistance and it is quicker running a Duro. Your results may vary.
Everyone has a favourite brand of tire typically, in a 700x38c, mine is the Cordoba, great grip, great traction, and great road speed.
Now on to
@olderthandirt on your question about the 26" 559 tires, it is going to depend on the Schwinn rim you are using, I know the Typhoon Tread is a regular 26 x 2.125 559 tire and works on many bicycles. Good luck on finding the tires that work, what you are doing is a great mission for a good cause.