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...