Was your transect data (the line) derived from some information that identifies the start and end location with coordinates? If you can ensure the line directionality is consistent with the direction the surveys were conducted (check a few). This isn't that bad... relatively..in any method I can think of you will have to manually verify directionality on each transect... or trust in probability and verify some subset.
First join the line feature to a table of the cross-section total lengths. These should be the same as the geometery length of the feature. If they are not you will need to figure out why. Sometimes distance measurements represent paths over the ground and are closer to 3D distances, but that should not be your case. The transects could go beyond the edges of the river though.
You will then need to convert your lines to routes using the Linear referencing toolbox (Create Routes) and ensure the directionality is correct in the measures.
Then you will make a route event (Right click on your routed line in the TOC)out of your length along transect/elevation table and export that to a point theme.
You can then use the points with the elevation value to convert your transect lines to a 3D feature. I can think of a couple of ways, but the quick and dirty would just be to make a TIN out of the 3D points and run the transect lines through the Interpolate Shape tool with this TIN.