I shave with a double edge safety razor, the kind your grandpa used to use. This offers a cleaner shave with less cuts, provided you know how to do it. It takes a little longer to learn but once you learn it is a much better shave and once you have some practice it doesn't take much longer.
This style of shaving requires you to mix shaving soap in a bowl or cup to create a lather before applying it to your face with a brush. In the past I used a coffee mug and then switched to an old shaving soap container. A few years ago I replaced it with this chrome shave cup.
I really like the cup, it is a lot smaller and takes less room when not being used. It doesn't rust or break down. I don't usually dry it out, I just give it a rinse and quick shake so it doesn't go away completely dry. If I was to guess, I probably had it for two years now and looks exactly as it did the day I bought it.
It is a great size for a shaving brush and allows you to quickly lather while wasting less shaving soap. It is cheap, only $6.99 shipped with Amazon Prime.
I switched from using disposable razors to using a double edge many years after reading a post on The Art of Manliness blog. If you are curious about the method and benefits, check out the post from 2008 that converted me.