There's nothing wrong at all with only lathering in sections. That can help a lot starting out, until you're able to shave faster. Since you're not able to shave right after a shower, try the hot towel treatment. While your brush is soaking wet a hand towel with hot water & hold it over your face. Depending on towel size, you might be able to clothes pin the corners behind your head to free up your hands to start lathering. It also sometimes helps after that to lather up your face, massage it in with your fingers & then apply another towel. When that one cools after a couple minutes lather up & shave. Your whiskers should be noticeably softer and easier to shave.

Edit: my water is pretty hard too. It helps to add a few drops of glycerin in your bowl when you start building the lather. That'll help it foam up better & stay wet a little bit longer.