House Guard Sliding Patio Door Handle Set, Mortise Lock, 3-15/16", Black | Fits 1-1/4" to 2" thick sliding glass doors; reversible design. Key cylinder not included.

You'll most likely need to drill new holes for the handles but it's always hit and miss with these old sliding doors. Something like this may work. Most mortis screw holes on center will be 3 15/16 which is very close to what you measured. https://www.amazon.com/dp/B09XJ8XT1F