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If AZ is out by a huge amount (like 20+ degrees as you have described) then you probably wont be able to make the required adjustment with the AZ bolts (with the stock bolts I think you have about +/- 10 degrees of movement max). In this case you'll need to rotate the entire tripod and mount setup and start the alignment routine again. When I was starting out, I was able to do this with the whole mount loaded up with telescope and counterweights. As others have said, using the solar noon method of finding south is an easy way to get the mount lined up fairly close to south.
I can also vouch for Alignmaster. After a couple of iterations of the Alignmaster software routine I typically have my error down to less than 1 arcmin (and often less than 20-30 arcsecs). Painless and only takes 15-20mins or so.
Once you have completed your setup the first time, it helps if you can mark the ground so that you can place your tripod in the ballpark next time. I have set a few pavers into my lawn, and use texta marks to align the tripod legs (see attached). By doing this i find that I can place the mount on the ground to within about 20arcmins in AZ and 5 arcmins in ALT before i even get started with the alignment procedure. certainly helps to speed things up.
btw, re your observation that moving the mount in AZ also moved the ALT by 4 degrees, if this is the case then your mount wasn't level in the first place.
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