Yes spaceships do have toilets to allow the astronauts the ability to use the restroom without the need of diapers.
Although astronauts sometimes do use adult disposable diapers during takeoff when they have to be strapped into the seat and cannot use the restroom.
But after they get into space they can then remove the diaper and use the toilet as normal instead of having to use diapers in space all the time.
The astronauts strap themselves to the toilet to prevent them from floating away while using the toilet and the toilet uses a device similar to a vacuum cleaner to suck away the waste that goes into the toilet so that it doesn't float around in space.
The astronauts also have showers in spaceships but the water droplets and soap float around and do not go down a drain.
However some space ships do not have any wash basins or showers but I've heard that a few types of spaceships do.