What can I do with like-new, recent-edition textbooks that I'm not allowed to sell?

Ask your employer.

You should be able to work out a solution both parties are happy with pretty easily. The obvious one is to sell the books and then give your employer the profits. If they don't care, you can also pocket the profits yourself, perhaps recycle the books or donate them to your university's library.

As an academic librarian, I am frequently asked similar questions. First, please don't just drop them off at the library. Unwanted donations are a significant problem at libraries—it's very difficult to recycle books, so libraries end up having to pay to get rid of books we don't need, on top of the time and effort it takes to deal with a big pile of books on our doorstep.

Do feel free to contact a librarian to see if they would be interested, but be prepared for the answer to be no: textbooks, which change edition quickly and tend to be most useful only to people taking specific classes, are fairly low on the list for most libraries. Some large university libraries do have policies of obtaining a copy of every required textbook, but they will already have a mechanism for purchasing those in a timely manner. If you do go this route, please also be as tentative as possible in your offer: Librarians also tend to dislike telling people "no", so if you push, the librarian is likely to take your books and then quietly dumpster them.

There are some charities that take donations of (some) books. Our library uses Better World Books, which I believe is currently accepting recent textbooks, and you could also look around for programs close to your home.

The most straightforward option might be to pass the books on to another student or students in your program. If you want your donation to have an impact beyond just the student who receives them you could ask them to "pay it forward" by passing along some other books at some future date. (You could also just ask that they pass your books along gratis, but the value of this gift will decrease with every iteration as the books get older and new editions come out.)

If you don't care about keeping any value within academia, you could offer the books for pick-up through something like Craig's List. Whether people want your books for study or a book art program or as kindling for their wood-burning stove, the books will at least have value beyond gathering dust in a box in your closet.

If you want to recycle some or all of your books, you will need to deconstruct them first, or locate a service that specializes in books (these are very hard to find, and may have a charge). If you want to do it yourself, a box cutter or utility knife to cut the pages out works pretty well. The pages can go to any paper recycling service. The cover itself may also be recyclable, but any glue and/or stitching will probably need to go into the trash. You probably shouldn't compost the books, as page coatings and inks can be toxic.

Give them to the professor. It is often useful to be able to lend the course book to future students who cannot afford the books.

Seeing if the library will take them is also a good choice, as mentioned in this answer, but they may not take them as mentioned here. Overall I think the professor can do the most good with them. He can lend the to the library, or sometimes the school office, for short term loans, give them to students for an entire semester, or decide that the library is the best place and donate them.