PRAW allows extracting submissions on a given subreddit between two timestamps using this:
reddit.subreddit('news').submissions(startStamp, endStamp)
However, I haven't been able to find anything similar for extracting a given user's comments between two timestamps. Can this be done? I don't actually care about the 1000-requests limit unless the comments I get belong to the correct time range. I already had a look at their documentation here.
Considering referring to Pushshift. You can get the comments by a user (let's say /u/avi8tr) at the following URL: Link .
There's a python wrapper (just like PRAW) for Pushshift as well, but it's under development: GitHub Link. You'll have to add the 'author' parameter in comment_search in psraw/endpoints.py, though.
Note: Both Pushshift and PSRAW are begin developed actively. So changes are expected.