What are these forms: かけちゃお, つないじゃお?

Question

I'm trying to understand the lyrics of a closing song of 「崖の上のポニョ」.

First verse is simple, but the second stopped me:

ペタペタ　ピョーンピョン
足っていいな　かけちゃお
ニギニギ　ブンブン
お手てはいいな　つないじゃお

What happened to arms and legs?

Answer 1

They're contracted from かけちゃおう and つないじゃおう, which are colloquial versions of 駆{か}けてしまおう and 繋{つな}いでしまおう, "let's run" and "lets connect", in this case 手を繋ぐ, "hold hands"

The auxiliary verb しまう usually means "do something accidentally", but in this case in the volitional form, it's used to express carefreeness.

手を繋いじゃおう Let's hold hands (and not care about the consequences)

A similar use is in the imperative

やっちゃえ！　Do it (and worry later)!

Answer 2

かけちゃお = かけてしまおう

つないじゃお = つないでしまおう