Learn English – “Before last year” vs “the year before last”

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.

Your grammar book is wrong: the present perfect is a present tense and is not used with temporal adjuncts which do not include the present. Last year without a preposition or a definite article designates the year before the current year, so it excludes the present, and you should not write

^∗Lots of people have helped us last year.

It would however be OK to write this:

Lots of people have helped us since last year.

And the last year is the year-long timespan which runs right up to the present moment, so this would be OK:

Lots of people have helped us during the last year,

and this, omitting the preposition, would be natural in speech:

Lots of people have helped us the last year.