How can I install modern locksets on these interior doors?
You're going to have to drill a larger hole for the new handles, which should completely encompass the old hole.

Head on down to your local hardware store (or cruse the internet), and look for a door handle installation jig like this Irwin® Door Lock Installation Kit


Example only, these jigs are available from many different manufacturers

It comes with the proper size hole saws to set up a door for modern locksets, and makes alignment of the holes quick and easy.  Simply attach the jig where you want it, and drill the holes with the supplied hole saws.