If you want to avoid a sweaty back, don't use a backpack. Having said that, you'll need a bike that can accomodate a rear rack or a front basket, or both. A hybrid can do that plus give you a more upright riding position so you're not putting too much weight on your wrists. A touring bike will also give you that, but you'll be bent forward more. Both can be fitted with fenders so you can safely ride through puddles with a minimum of mess.