Based only on that PNG image. I made this file primarily using QGIS, and the following public domain [
https://www.census.gov/geo/maps-data/data/tiger-cart-boundary.html shapefiles from census.gov]:
cb_2015_us_state_20m
cb_2015_us_county_5m
The map projection is:
NAD83(2011) / Texas Centric Albers Equal Area
+proj=aea +lat_1=27.5 +lat_2=35 +lat_0=18 +lon_0=-100 +x_0=1500000 +y_0=6000000 +ellps=GRS80 +units=m +no_defs
The original PNG, rotated and scaled slightly to fit this version of the Albers EA projection, is in the image for your comparison.
The approximated areas of the Dust Bowl were traced from the PNG in InkScape, no shapefile was made or used.

This is definitely the current crown jewel of my Wikipedia images. It took the longest to research and make and it took lots of software. But it was definitely worth it - this map has been requested by readers of w:en:Dust Bowl three times. Someone had to deliver eventually.
Based on map by ID m12545a/55 by 
Natural Resources Conservation Service, 
Soil Science and Resource Assessment, 
Resource Assessment Division (NRCS SSRA-
RAD) Baitsville, MD Jan 2012
The source of that map:
Map 2579, March 1954,
U.S. Dept. Of Agriculture
Soil Conservation Service

https://www.nrcs.usda.gov/wps/portal/nrcs/detail/national/about/history/?cid=stelprdb1049437
Soil Science and Resource Assessment, Resource Assessment Division (NRCS SSRA-RAD) (Division of the U.S. Dept. Of Agriculture)