They already have a process for bulk signing. Is there a reason you can't use it?
A convenient feature is the creation of templates. If you need to sign documents in a bulk mode, create a blank by adding your name, title, and other necessary data that are repeated in each document.
"eSign Documents Free - Online eSignature Service - DigiSigner" https://www.digisigner.com/free-electronic-signature/esign-document
There are dozens of electronic signature solutions on the online market and each one offers different services which you or your company may benefit from.
There’s another thread in Quora with a full list of the best ones here.
DigiSigner is a great option to start with as it offers a free trial of the full service and also 100% free software version with no registration required.
Free Version: Sign PDF Online with DigiSigner
Full Version: Free Electronic Signature Service
DigiSigner is used by companies of all sizes and across numerous industries, such as real estate, legal, HR, and more. Although it is a basic tool, it works effectively and helps you avoid wasting time and other resources. The digital document signing company offers a free account.
This one could definitely be included in that article as they did in this article from entrepreneur.com