Can I change the email address shown in the e-signature emails I send to clients?

If support can not help, that means the information must be changed on your EFIN and you need to contact IRS https://www.irs.gov/tax-professionals/how-to-maintain-monitor-and-protect-your-efin

When done you will need to submit EFIN to intuit AGAIN. https://proconnect.intuit.com/support/en-us/help-article/account-management/register-efin-proconnect-tax/L9UQWErbz_US_en_US?uid=ldsziicn

EDIT - I stand corrected by Phoebe, who knows way more than me. Ignore my post above.

I assumed you clicked the Settings "gear" in the lower left and made the change there Preparer Information

New guess!  - https://proconnect.intuit.com/support/en-us/help-article/accounting-bookkeeping/managing-intuit-account/L3EwuDVOC_US_en_US?uid=ldt00g91

change your e-mail there. 

It really sucks when support doesn't know how to do such a mundane function.

I have been trying to fix this for a long time, and finally stumbled upon the solution!!

To change the e-mail address shown in the e-signature requests, you need to log into your Docusign account linked to the 'incorrect' email address.  Then go to Manage Profile in settings, and update to the correct email address. 

There was a bit of a verification loop to go through, but it updated rather quickly.  I sent a test e-signature request to myself, and it included the correct/updated email address.  The content of my e-signature dashboard in ProSeries was not impacted, all pending and completed documents are still there.