Yeah, only left without giving notice once. About 22 years ago. Working for an electrical supply firm doing multi drop deliveries for them to building sites all over within the M25 (Cable/Conduit, that type of stuff) Needed a job after uni while applying to music companies for a job. One came up after a year of knockbacks, so I interviewed for it on a Wednesday, they offered me the job Thursday to start on Monday.
So on the Friday morning I walked into the branch managers office and told him today was my last day. He took it very well as he knew its ultimately what i wanted to do and shook my hand and sent me on my way and said there is always a job there for me if it didn't work out, he was a decent bloke and a great manager. I think that was a nice display of the effort I'd put in working there for a year which they'd seen despite it not being my choice of career which they knew. I always had the ethos of if you're doing something and they're paying you, do it well.
Oddly enough though, before I went to his office to tell him I was leaving, one of the company sales reps came to speak to me about moving into the office on sales and getting out of the warehouse, I just told him I was on my way to see the gaffer to tell him I'm leaving at end of the day, he couldn't believe it!
With regard to your dilemma. They clearly don't value you where you are or they'd have tried to do something to keep you, they're just being petty over a week. If it was a month's notice it'd probably be fair enough. Tell them the new position is better terms and you like the job, so you're going. They won't do anything about it despite it being in your contract, too much of a ballache for them to take any action.
