CF for expiry year only

Welcome to the forum.

I'm not 100% from your question exactly what you want or the full extent of your problem, but I think part of it is that you have 'formatted that to show the YEAR only'. Formatting in Excel doesn't change the base data, only the way in which you view it. So when you format a date to show the year only, the underlying date is still day-month-year. (Actually, in Excel a date is just a number, with 1 being 1st Jan 1900 - but that's not relevant to this point.)

If you want to check against the year only, you need to convert your dates to just years. Luckily you can do this easily with the YEAR function.
=YEAR(TODAY()) will give you the year for today (2021 right now).
=YEAR(A1) will give you the year of the date in A1 (assuming there's a date in A1).

So, let's say you have a date from 2021 which is stored in A1 and you want it to show as near expiry when you reach 1st Jan 2026 because that's nearly 5 years later, you can use this as your CF formula:
=YEAR(TODAY())=YEAR(A1)+5
For past expiry, just replace the second = sign with > instead:
=YEAR(TODAY())>YEAR(A1)+5


You also note that you'd prefer 3 months to 'next month'. Try using EDATE or EOMONTH for this.
With your date in A1, use this CF formula to give you a check if today is 3 months or more later:
=TODAY()>=EDATE(A1,3)
To check if today is in that third month, you could use this:
=TODAY()>EOMONTH(A1,2)

Give those a go and get back to us if you need more explanation or help.