Nearest neighbor distance in km over contiguous US

I have two sets of line features in QGIS (rivers & roads – I’d like to find the road segment nearest to each river segment & the distance between them in km) across the contiguous United States. I’ve tried using the nearest neighbor join plugin (NNJoin) with both WGS 84 and the USA contiguous equidistant conic (ESRI: 102005) projections, but I get distances between 0 & 50, which I’m assuming is in degrees. Should I try a different projection or is there a way to manipulate the NNJoin plugin?

Edit: I have also tried keeping the river segments in WGS 84 & only putting the road segments (join layer) in a projected CRS (as per the help page on the plugin). However, this reprojects the road layer to a totally different location from the river feature & also gives clearly incorrect distance values.