There are many, many verbs in English that describe walking movement. A search using the Advanced Search feature of the Macmillan English Dictionary CD-ROM gives over 60 results, including verbs such as patter, waddle, totter, shuffle and stalk, but none of them seems to fit this particular case. I don’t think there is a specific verb in English to describe this movement. The best verb I can think of to describe the way birds walk is bob. This is generally used for all kinds of up-and-down movement, such as boats on water, or someone’s head:

The little boat bobbed up and down on the waves.
She turned to her sisters now, her head bobbing.

It’s quite possible that this verb came to mind because of the words of the old song about a robin that is ‘bobbin’’ along. Of course for the song, the verb was probably suggested by the rhyme with ‘robin’, but it still seems a pretty good way of describing a bird’s movement, back and forth as well as up and down:

A common sandpiper bobbed on a boulder on the hillside.

Why is it that tanorexic means someone wanting a tan, yet the word anorexic means (in effect) going without food. The ex contained within the words comes from the Latin, meaning ‘without’. So surely tanorexic would mean not wanting to tan, and tanaholic would refer to someone who tans a lot? Perhaps you could shed some light on this.