The library policies may change depending on what the vendors offer. My library has Overdrive (X number of copies, wait till one becomes available as with traditional books) but also Hoopla, which limits you to 8 items per month but does not seem to limit number of copies of any given item lent at one time. The issue with Hoopla is that books don't necessarily stay available (similar to Netflix, where movies come and go). I hit my limit one month and spotted some other things that looked good--by the time I was eligible to borrow again, they were gone! (I have read that some of the Netflix-like e-book services have had to cut down on the number of romances they offer because the readers go through them so fast it was messing up the financials they had projected.)
Back to the original poster--Amazon does have daily and monthly sales on ebooks (there's an email you can sign up for.) A lot of it is genre fiction by lesser-known authors that the libraries are likely to have also but every once in awhile they will have some variety--Terry Pratchett's books have come up from time to time, for instance, and today one of the books was the classic (but not classic enough to be free) memoir Cider with Rosie. I'm not sure what your budget is for books but in a lot of cases a used hard copy (even with shipping) is cheaper than the e-book if you are willing to wait awhile.
Overdrive and Hoopla have different pricing structures for libraries. Hoopla charges libraries for each checkout (varies by media) - my system just began using Hoopla, and we've set up a base pool of credit for checkouts, with provisions for future overage charges. We shall see how it plays out budget-wise after a full year.
Overdrive allows library systems (and individual libraries) to purchase the right to circulate individual copies of ebooks. Unlike a physical book, these ebooks have a fixed number of circs permitted (26, I've heard) and then it goes poof. A physical book can be circed hundreds of times. As Rubybeth mentioned, libraries pay an exorbitant amount per ebook copy (vs individual purchasers). Certain publishers refuse to sell ebook rights to libraries AT ALL, so some bestsellers will never be available thru libraries. We are terribly frustrated by these restrictions from the publishers.
Personally, I periodically check Amazon's top 100 free list. Lots of self published stuff, but occasional promotional freebies of a known author's first in a series, to attract new readers. It changes frequently, so I check often - the promotions may be one day only.
Out of copyright classics can be free (Gutenberg Press, Amazon), but can also be a "new" edition that is not free - check carefully.
Overdrive, Hoopla, Amazon free books are all I've tried to date.
Regarding Amazon Prime - I've got a kindle ereader, a kindle fire, and DS5 has a kindle fire. All are linked to (registered to) the Amazon account in DH's name. I've renamed each device to distinguish them when downloading things. Is your kindle registered to your husband's Amazon account, or do you have separate accounts?