I am aware that the pairing for state goes as follows:
Dist.1 vs Dist. 2
Dist. 3 vs Dist. 4
But can anyone shed light on how that plays out exactly in the 16-man bracket? Thanks for any help anyone can give.

I believe they mean the 16-man bracket lines up the four No. 1 seeds against their opposing District. For example, the #1 seed from District 1 (1A) wrestles the #4 seed from District 2 (2D), #2 seed from District 2 (2B) wrestles the #3 seed from District 1 (1C), and so on.

1A vs 2D
1B vs 2C

1C vs 2B
1D vs 2A

3A vs 4D
3B vs 4C

3C vs 4B
3D vs 4A

Don't make any assumptions - the OFFICIAL state bracket will not be posted until Sunday AFTER District Tournaments -

Leanna Grater
State Tournament Committee

and for good reason too, that way people don't rig matches to get "better draws".