use 'trynewoed" as id/password

Despite your subject title, I read the password first as tryne-woad.
Dare I ask how much it costs?

you mean after the "free trial" is over? you don't want to know - I think the value of the trial is to see if it's worthwhile to you to figure out how to connect through your library, which should be no cost to you. (I used to have to log in with my 37-digit [hyperbole] library card no.; now it's a direct link to the OED home page through the library's proxy.]

Ruh roh--I thought I might be able to do it--there is a link on our library's web site. But after I put in my card #, the OED site gave me this:
To proceed with authentication to this resource, please select your library or institution from the following list:

Buffalo & Erie County Public Library

Manukau Libraries

I do not live in Buffalo or Erie County, and I have NO idea what or where Manukau is. I may inquire further when the library is actually open, and get a better outcome.

yes indeed, if your library's web site has a link, ask them how to get connected!

edit: things may have changed since the relaunch; I had to notify HCL to get our library to update their link!

Manukau Libraries

Could be the library in my hometown.