You don't really need a nested formula, if your date is in A1 try this to give the period, works regardless of year

=LOOKUP(TEXT(A1,"mmdd"),{"0101","0501","0725", "0906","1025";"A","B","C","D","A"})

If your formula shows as text probably the column is text-formatted - to convert try this:

Select column and use Data > Text to Columns > Finish