The way you define worship would imply a life that is in whole an expression of worship of God. And, in fact, that's how it should ideally be. But without any fanfares whatsoever. By public worship I mean putting on a display of worship: specifically, prayer and/or fasting (or similar), but also all rituals, such as going to church, and basically anything done as an outward expression of worship -- now I worship, now I live my life. True worship of God is a private affair -- to an outside observer you just appear to simply live your life while inwardly you know what's the real deal. After all, worship simply means "to acknowledge worth."

In other words, no outward behavior automatically means worship, but one's private attitude of giving worth can turn all outward behavior into worship.

As for Shabbat -- yes, worship, rest, or service certainly don't break it per se. But neither does work or just caring for oneself. You don't observe Shabbat by adhering to some prescribed set of actions or lack of them. You observe Shabbat simply by unconcerned being and appreciation of all what is, no matter what you do or don't do. Yes, rest is conductive to that, but just being at rest (on a particular day of week) does not mean you automatically observe Shabbat. It's basically the same case as with worship.