I want to reuse, in a new work, some period drawings and paintings that are held in various libraries and archives. Presumably this means paying for their time spent helping me. To make the process go as smoothly as possible, what is the correct process for obtaining high-quality copies of these materials and permission to republish them? Will I need to already be working with a publisher before acquiring the artwork? Will I need to sign a terms-of-use contract with each archive? Will I be allowed to crop and adjust color levels to improve print quality? Is there anything in particular to watch out for?
Before requesting publication rights for images it is best to have an arrangement with a publisher, as most institutions require you to provide details about the publication venue, the type of publication (print or digital or both), the print run and so forth when you are requesting the images. Then you have to make a request to the image holder (provided the copyright is not an issue for your, as you indicated in a comment to another answer) either using a web form or by email (in most cases you should be able to figure out from their website who the contact persons for such requests is. They will inform you about the costs, and then you can decide if you go on with signing an agreement (yes, you have to sign a contract with each institution providing images even in the rare cases when the images are provided for free), and they are expected to provide you with high resolution images after the payment (if they charge a fee, as most such institutions do, at least in my field). You surely can crop and process the images as required for the production of your publication. Keep in mind that in the worst case, the process can take months, so it is best to start it as soon as you reach an agreement with the publisher.
Any creative work is either owned by its copyright owners, or is in the public domain.
When it is in the public domain, you can reproduce and alter the copies anyway you like.
When there is a copyright owner (and from your question I would suspect that this is the case), then most reuses of the creative work requires permission of the copyright owner. Some copyright owners will be very liberal in what they allow you to do, while others will be very restrictive. Some will demand money, others won't.
So, the first step is finding out who the copyright owner is. Then the second step is to propose your plans with the creative work to them, and see whether they allow this and whether they require any (monetary) compensation.