Sung by Ruth Etting in 1928 and by Helen Kane in 1929 (the big hit of the time). Both on youtube.com.
Many small variations in on line lyrics and order of verses (I haven't seen the sheet music).

Helen Kane lyrics:

Lyr. Add: Button Up Your Overcoat

Listen, big boy,
Now that you got me made
Goodness, but I'm afraid
Something's going to happen to you!

Listen, big boy
You gotta be hooked, and how!
I would die if I should lose you now!

Button up your overcoat
When the wind is free
Take good care of yourself
You belong to me!

Eat an apple every day
Get to bed by three
Take good care of yourself
You belong to me!

Be careful crossing streets
Cut out sweets, ooh-ooh!
Lay off meats, ooh-ooh!
You'll get a pain and ruin your tum-tum!

Wear your flannel underwear
When you climb a tree
Oh, take good care of yourself
You belong to me!

Button up your overcoat
When the wind is free
Oh, take good care of yourself
You belong to me!
(Boop-boop-a-doop!)

When you see a traffic cop
Use diplomacy
Just take good care of yourself
You belong to me!

Beware of frozen ponds, ooh-ooh!
Stocks and bonds, ooh-ooh!
Peroxide blondes, ooh-ooh!
You'll get a pain and ruin your bankroll !

Keep the spoon out of your cup
When you drink your tea
Oh, take good care of yourself
You belong to me!

Don't sit on hornet's tails,* ooh-ooh!
Or on nails, ooh-ooh!
Or third rails, ooh-ooh!
You'll get a pain and ruin your tum-tum!

Keep away from bootleg hooch
When you're on a spree
Oh, take good care of yourself
You belong to me!

* thinking of scorpions?

Mostly from lyricsfreak.com