I'd better go now or I'll be late.

Hello!

Can I use should instead of had better?

*I have to meet Amy in ten minutes. I'd better go now or I will be late.*

I've read that should is used in general situations while had better is used in specific situations. Is that right? Because other example that I saw

You look tired. You should go to bed.

I thought this was a specific situation, but they use should in this example.

Thanks in advance.