8/23/32 the Wyoming State Highway Commission approved US 16-N to replace US 116 from Lovell to Ucross via Sheridan and US 16-S to replace US 16 from Lovell to Ucross via Buffalo. See pg. 26 in the 3rd pdf document at https://drive.google.com/drive/folders/1kmrrq5P2bx3RQgkhu-JXDBWABUV85b_X
Pg. 91 of the same document from 7/27/33 shows them replacing US 116 with US 14 (along with US 216 becoming US 16 and US 420 becoming US 14) with no mention of US 16-N or 16-S.
So it is unlikely US 16N and US 16S were posted in the field. Appearances on a map would likely be limited to 1933 issues if at all...
All Wyoming Highway Commissioner Minutes from 1917-86 are available online - https://rmoa.unm.edu/docviewer.php?docId=wy-arrg0045.xml#idm954240
The original assignment of state highway mileage is listed very specifically in 11/25/29, 1st pdf document pp. 1-11 at https://drive.google.com/drive/folders/1kmrrq5P2bx3RQgkhu-JXDBWABUV85b_X - no state highway numbers are referenced through at least Oct 1936
US 16N and US 16S? This is the first time I've ever heard of this. Even the defunct US Highways 1-830 page never mentioned there was to be a US 16N and a US 16S in Wyoming. Very interesting.