I would try using a lamp auger bit on the lathe as a pilot hole then follow with the 1".  Drill as deep as you can before using the extension. 

Matt, odds are strong that your forstner did not start true if it "wandered".   That is because forstner bits guide on the outside rim, and are designed to follow the cylinder created. 

Best protocol with forstner bits and long holes is to start the hole with out the extension and after establishing the cylinder, adding the extension. 

A fluted bit seeks to center on the center point, but differences in grain, knots, etc can cause the point to deflect ( but not so much with 1 inch bits). 

I like forstners for the smooth holes in boxes and peppermills.  They do need to be sharp and the rim free of dings.

One option for long bits is morse tapered bits that fit directly into lathes or drill presses.  Larger diameters are costly with a 1 inch bit about $60

I use Colt bits with the colt mt taper adapter and extension for pepermills with good success.

Matt, you might want to read this thread.
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We covered much of the same information your looking into.

Matt, the stepped bits I have are designed to work on thin materials like sheet metal as the steps are short.  None of those I saw browsing Amazon had a max diameter of an inch. 

I think I would treat clearing out a small amount of wood like an end grain box.  I tend towards Hunter cup cutters, but a small gouge and a smaller square scraper would do nicely (and quickly). 

Bonnie Klein and Ray Key are two great box turners who use gouge and small acrapers.