I haven't used the Suzuki books. I thought they were for the Suzuki method though -- emphasis on memorizing pieces and not reading until you can play. Although... they do have to learn to read sometime... maybe you have one of those type of books. I just haven't seen them. I've seen some for violin, meant for pieces being memorized.
If you don't look at your hands, fine. Probably a good thing. If you need to look down for a jump now and then, that makes sense. Just don't be too tied up on not looking at your hands. Actually, if you're looking at the music, I would stop worrying about it. Focus on other things then with your attention.
Fingerings -- Use what they have. Make your own. Either way. After awhile it will become second nature. Unless it's actually the composer, if it's just an editor, you can use what you want. Even if it is the composer, your hands are your hands, so use what you want. Again, I wouldn't worry too much if you're using their fingerings. I let my hands figure out what's right for them.
I generally lean toward the idea that people taking traditional classical lessons aren't sight-reading their music. They work on it, using the score. At first, pieces take a week. Then some a month. Some six months. If you're sight-reading though, that's great. The way I was taught it wasn't emphasized much.
On that note, if you wanted to sight-read... Why not get a few different series, complete series, method books and go through those? You'll hit all the basics. If you know it already, you can sight-read through it. By the time you get to the end of those books, you'll be playing pieces.
I know Alfred (I think) has an adult beginner series. You can bet the others do too. Faber & Faber. Bastien. Do a search and you'll find lots of publishers. I've never been impressed much with anything I've seen from Hal Leonard or Yamaha for method books. But that's just me.