I Googled "adoption privacy laws Massachusetts" just as an example. Put in your state when you search.
This was about the 4th or 5th hit on the page. Results may vary depending what state you put in the search term.
Adopted Child's Right to Information as to Biological Parents
https://www.stimmel-law.com/en/articles/adopted-childs-right-information-biological-parentsThe article is completely about the child's right to find their biological parents, not the other way around. And it comes from one law office, not from "on high", so it might be slanted or at least not the source of absolute truth on the matter. Look for a few other opinions before coming to a conclusion.
Then I Googled "biological parents rights massachusetts" and found this:
Massachusetts law about adoption:
https://www.mass.gov/info-details/massachusetts-law-about-adoptionYou will probably find a similar page for your state.
I am starting to think that neither you, nor your mother, will be able to find your half-sister through the records. She can probably find your mother, or some information about her, but not the other way around. I might be wrong.
Which brings you back to DNA. Watch for a sale on autosomal or "family finder" DNA. For example, familytreeDNA is having one now until April 25,
https://www.irishgenealogynews.com/2019/04/dna-day-special-offers-from-family-tree.html[I am not connected to familytreeDNA.]
First find out if you will be able to upload the results to other sites like ancestry.com. That way you will be able to compare the results with more people. Or wait for ancestry.com to have a sale, and upload it to familytreeDNA. There probably have been rootschat threads that answer which sites will take other sites' results.
I think it would be better to submit a sample from your mother, if she is still alive, than from you. That way you will eliminate matches from your father's side. Ask someone else to be sure.
Then hope your half-sister has submitted a test as well. If so, it should be a close enough match to identify her to you and you to her.
Edited to add: I just noticed that Shelleyesq suggested ancestry.com, since it has the biggest database.
Of course, for $49 apiece on sale, you might do tests on a couple sites.