Integrate does not allow you to only move one dataset. Despite any weighting you do, in some situations one dataset will eventually move.
One way of getting around this is to set an aggressive threshold distance for the integrate, then use the spatial join tool with the 'join at line segments' option to spatially join the two road networks. Before you begin, ensure that you have a backup of both datasets, and that the dataset you want consider the primary one has an ID or primary key field, like FID or OBJECTID.
Because of the integrate, all the line segments that somewhat overlap should now share at last two vertices, allowing the spatial join by segments to work correctly.
Once the two datasets that have been integrated have been spatially joined by line segments, you should then have an output dataset that contains the attributes of both datasets, but the geometry was probably mangled by the integrate. However, the dataset that you didn't want to move should have an ID field that was not changed by any of the preceding processes. You can then take the dataset you just created, and the ID field that it contains from the original dataset, and join that by attributes (the ID field) back to the backup, unaltered, version of your primary dataset.