The limit they are talking about is 50% of your adjusted gross income (AGI). (for donations to public charities. some private foundation contributions are limited to 30% of AGI). (For 2020 the limit is 100% of AGI because of the pandemic CARES Act.)
So if you have an AGI of $40k and make contributions to a public charity of $30k in that year (except 2020), your deduction is limited to $20k (50% of $40k). You get a $10k carryover.
If in that year you took the standard deduction, that means you did not actually take an itemized deduction for the $20k of allowable deduction. What I quoted says you can still use the $10k carryover in later tax years.
This probably doesn't apply to you. It doesn't seem likely that $2k annual contributions will be over your AGI for those years. Your AGI would need to be less than $4k in those years, which doesn't seem likely. In that case you wouldn't need to file and probably wouldn't have any tax liability.
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