For someone who's been around as long as you have, you should know that's not a table.  If people can't recreate your scenario, you're not likely to get an answer. ...

What do you mean by "nearest to today"? Can be either before or after?  Then you'd have to do MIN of ABS(DATEDIFF....)  or return the top 1 value...

You shouldn't be grouping by EventDate - you're trying to return a MIN([EventDate]), so grouping by it (return each unique value) doesn't make sense.

If you have control over the design of the table, you could create a new column and extract just that part to the new column. Then you could index it...

Could you post some sample data - insert statements so we have something to work with? Basically, you can use LAG with a window to get the "previous" record, and...

Maybe this?
SELECT member,
    amt_paid,
    amt_adjusted,
    dt,
    dupeNum
FROM
( SELECT member,
    amt_paid,
    amt_adjusted,
    dt,
    ROW_NUMBER() OVER (PARTITION BY member, dt, amt_adjusted ORDER BY member, dt) AS dupeNum

Kinda would help if I wrote an actual delete query... =)
Test the CTE out by doing SELECT from it first instead of DELETE
;WITH
    cteMembers(member, amt_paid,...

Like this? It's not super clear what you want. In this case, a sample expected result would be a big help.  Something like this?
Create table #t( member...