How to get text to warp how I want it?

Hey all I'm struggling to get my text to warp right.  I'm creating an effect of typing on a takeaway coffee cup.

I've got the typing bit done.  I'm using warp to get the writing to look like it's written on the cup.

But using warp I can't get it to look right.

If I really increase warp amound the text goes off the screen and I can't warp it anymore.

Why is text disappearing when the red dotted box goes of screen?

I hope this is making sense?

Or if anyone knows a better way of doing this I'm open!

I've included images which I hope explains the issue better.