As Tracy R. wrote in her recent column entry, 'should' is more relaxed than 'had better': http://cafetalk.com/column/read/?id=14623&lang=en

In this column, I'd like to add some information about 'had better': 

When you hear Americans using 'had better,' you will almost never hear them actually saying 'had better.' Usually, they will shorten this to 'better.'

Example: You better go see a doctor (instead of 'You had better go see a doctor') 

When I was a kid growing up in the U.S., I ONLY knew the short form. It wasn't until I was in high school that I learned that this was supposed to be 'had better.' Even now, I only use the short form when I speak English. I think most Americans are the same way. Do most British English speakers do this? I'm not sure. 

If you want to hear examples of this (had better → better) in a TV drama, take my TV DRAMA lesson!