Projection problem for Chinese river file into WGS84 43N

I am using QGIS 3.10 and having issues with (re)projection of two shapefiles. For a map I am trying to make of a river network in Xinjiang China and South East Kazakhstan, I have found two shapefiles of Chinese rivers.

These were originally uploaded to the site in "Albers Conic Equal Area, datum WGS84". My project is in WGS84 43N. Adding the layers initially gave the error of "invalid CRS".

Already tried reprojection, saving in different CRS, adding the shapefiles as a first layer in a new project – all to no avail. The shapefiles are now in the desired projection, yet they still hover above the South Indian Ocean.

Any tips on what I could try and do?