I'd still use the /questions method, and change the filter to include answer details. E.g. [this query][1] lists all Stack Apps answers in the support tag since April 14th. All I had to do was to edit the default filter and check the checkboxes for the fields 'answer' → 'body' and 'question' → 'answers'.

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

[1]: https://api.stackexchange.com/docs/questions#fromdate=2020-04-14&order=desc&sort=activity&tagged=support&filter=!-*jbN-(0_ynL&site=stackapps&run=true [2]: https://i.sstatic.net/KmFbS.png

I'd still use the /questions method, and change the filter to include answer details. E.g. [this query][1] lists all Stack Apps answers in the 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. 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}.

[![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