I'd still use the [/questions method](https://api.stackexchange.com/docs/questions), and change the [filter](https://api.stackexchange.com/docs/filters) to include answer details. E.g. [this query][1] lists all Stack Apps answers in the [tag:support] tag, newest first. All I had to do was to edit the default filter and check the checkboxes for the fields 'answer' → 'body' and 'question' → 'answers'.

There is no way to filter on answer date upfront, but if you sort by question activity, you can ignore older answers and stop searching when you see a question with a `last_activity_date` older than a week ago.

[![enter image description here][2]][2]

An alternative would be to use the [/search/excerpts method](https://api.stackexchange.com/docs/excerpt-search#fromdate=2020-04-14&order=desc&sort=activity&q=is%3Aa&tagged=support&filter=default&site=stackapps&run=true). This seems to work too, but it doesn't return the full body, so if you need that, you'll need to make another call to e.g. [/answers/{ids}](https://api.stackexchange.com/docs/answers-by-ids).

[![enter image description here][3]][3]


  [1]: https://api.stackexchange.com/docs/questions#?order=desc&sort=activity&tagged=support&filter=!-*jbN-(0_ynL&site=stackapps&run=true
  [2]: https://i.sstatic.net/KmFbS.png
  [3]: https://i.sstatic.net/lJJEV.png