I use an alcohol based after shave after every str8 shave. Some people with dry skin can't use alcohol based AS, but it has nothing to do with a str8. If you use alcohol based AS with a DE, you'll be fine with a str8.

When you first start shaving with a str8, your face is a little tender as it gets used to the blade (and as you adjust your technique to no pressure and low angle). You might want to use an AS balm until the tenderness goes away.