Learn English – “Dripping with sweat” versus “dripping sweat”

Away just means away, i.e., away from home, but doesn't necessarily mean out of town. You might have been staying in town at a friend's house. But it sounds like you really were out of town, and thus you can use either one. They are both very commonly used phrases.

Of course, if you really don't want to give any details you can simply answer:

No, last week I wasn't at home.

In looking into questions of synonymy (or its absence) it is important to keep in mind that individual words and phrases themselves do not have atomic meaning: they have not only ranges of meanings, and different meanings in different contexts, but also different ranges of meanings in different contexts. So even in such distinct contexts as you provide you may discover instances where the two phrases are essentially synonymous, instances where only one or the other phrases is usable, and instances where both may be used but they have different meanings. There is not necessarily a 'core meaning' to any word/phrase, or 'core difference' between two words/phrases, which translates predictably from one context to the next.

Let's look at the phrases 1) by Xself and 2) on X own in three different contexts: a) with do, b) with live, and c) with be.

1a: I did it by myself.
2a: I did it on my own.

• Both by myself and on my own may mean ‘without assistance from anybody else’. But by myself may also mean ‘when I was alone’; and on my own may also mean ‘on my own initiative, without being prompted or ordered to do so’.

1b: He lives by himself.
2b: He lives on his own.

• Both by himself and on his own here may mean simply ‘without any cohabitant’. But by himself may be used with an implication of isolation or loneliness, while on his own may be used with an implication of independence and self-sufficiency.

1c: We're by ourselves.
2c: We're on our own.

• Both by ourselves and on our own be used to mean ‘without assistance or support’, ‘left to our own devices’; but on our own may also express a sense of opportunity to exercise initiative. And by ourselves is more often used to signify privacy—“Alone at last!”—and consequent freedom to pursue more intimate or surreptitious activities.