I got some new wheels, with a SRAM 1070 cassette, 11-23. I had been using a SRAM 1050 11-25, and I could, and still can, shift into every gear.

On the new wheels, it WILL NOT shift into the 23, no matter what I do trying to adjust the derailer using the barrel adjuster, basically leaving me with a 9 speed 11-21.

Any suggestions?

If adjusting the cable tension isn't working, it could always be your limit screws on the rear derailleur. One of them controls how far in it can go (and also keeps the chain from flying into the spokes), the other controls how far out it can go (keeping it from flying into the stays).

The lower limit screw may need adjustment?

Limit screw.

Is the dish on the wheels the same or did you have to use spacers on one?

Screw limits.

To restate it in a different way, screws of limiting.

Test your limits. Think outside the screw.

limit screw for sure its the one with the L beside it. just turn it in small incremints until you can shift to your large cog. turn it as if you were losening the screw.

sounds like you need a new bike....

ftw

Once you make the adjustment, be careful if you switch back to your old wheels. You may need to re-adjust the limit screw to prevent shifting into the spokes. Or it may be such a small adjustment that there's no problem.

This post is usually followed by the post where the chain is thrown into the spokes with significant damage.

Go to Park Tool's website and download the thorough directions for adjusting the RD and FD. Then spend considerable time studying this so you learn to do it right (it isn't that intuitive). If you aren't inclined to educate yourself, take it to a LBS.

FTW +1
Now that is funny.
OP...new wheelset has smallest rear cog likely slightly outboard of old wheelset cassette smallest cog. You will also need to adjust your cable tension.
If your cable isn't sticking and your rear derailleur spring is still strong, it comes down to the small cog limit screw and/or cable tension.
To get the derailleur to shift more outboard you need to loosen the limit screw and/or loosen the cable.
Nothing to it really.

limit screw. do it carefully and not too much too quickly.