When I set up a guitar, I'll set the relief of the neck first, THEN adjust the saddle and finally the nut. I've fixed a a few guitars where it seemed the previous tech would change a saddle or nut and set the action WITHOUT checking the neck relief first; then when the neck is properly adjusted the saddle or nut ends up being too low and shims are used.
If the first four frets are buzzing it's possible that the neck is either too straight or maybe even slight upbow. If so, the saddle will have to be set higher to eliminate buzzing. Then if you drop tune, the neck will straighten out or backbow more (due to the decreased tension) and it will buzz more.
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