I donated 26 boxes of books, and the librarian estimated a worth of $19,500 in writing. Can I claim the whole value? "What appraisal method "do I choose?

[MKB5]

I also have no receipts as this is a collection made over 80 years.

[JosephS1]

The librarian's estimate of value should be acceptable provided they have prior experience in determining the value of donated books and they are willing to attest and sign Form 8283 for noncash donations in excess of $5,000.  The link to the IRS page pertaining to contributions is below and should assist you.

Charitable Organizations - Substantiating Noncash Contributions

@MKB5 

[Opus 17]

To claim a donation of non-cash items more than $5000, you need

1. A form 8283 signed by a qualified appraiser and signed by a qualified financial official of the organization that received the donation.

2. A signed written appraisal by a qualified appraiser, that must include a specific declaration required by the IRS. See page 20 here. https://www.irs.gov/pub/irs-pdf/p526.pdf

Who is a qualified appraiser is described on page 21.

Also, you can only claim the fair market value of the items if the organization keeps the items to use for their intended purpose.  In other words, the library must add these books to their collection.  If the library sells the books for a fundraiser, then you can only claim your cost basis, or the fair market value, whichever is lower.  Your cost basis is what you actually paid for the books.  If you don't have proof of your cost basis, and you are audited, the IRS can disallow the entire deduction.  

In turbotax, you can list the donation, and turbotax will prepare an electronic version of form 8283 so you can e-file.  However, after e-filing, you must mail the original signed form 8283 and the original signed appraisal to the IRS within 3 days, using a cover page (form 8453) that Turbotax will tell you to print.   Keep copies for your records.