I know it's tempting to change your Twitter and Facebook avatars, but when you have a business profile, it's necessary to keep your avatar the same. The best solution is to keep your avatar as a logo. If it's important to change it up, then I suggest you create a personal profile that you can change to reflect your personality. I for one recognize everyone on Twitter, Facebook and other social networking by their avatar. I have recently been running into the changing avatars for shops that I'm looking for and cannot find because the avatar has changed to the latest theme on Facebook or their pets latest pictures. I know, that's fun! I'm happy to see them, but tell me about them instead.
I know the handmade community like correspondence on the personal side, and you still can by using a consistent avatar. You can post in your Facebook photos tab your latest dog's trick or you child's first steps. Or blog about it. It makes for another example to draw people in and see all your latest products. Your customers and potential customers don't get bored with your logo or avatar. It's how they recognize you. They grow to know it and expect to see it. McDonald's golden arches do not change, ever and neither should your 'golden arches'.