you'll need to go through the same income verification as anyone else. If you have alimony payments, you'll need a copy of the court order which states the amount of the monthly payment.
I bought my current home five years ago after my divorce, and that's exactly what I had to do.
But you need to think long and hard about whether or not it makes financial sense to keep these properties. Can you really afford the payments? Can either one provide rental income?
I don't know for sure about your specific credit record. If you had credit cards in your own name, that will help, because you'll have your own payment history. Utilities in your name means you do have a payment history. It may be possible for him to sign a quit-claim deed to the properties, after which they'd be re-titled in your name, and you might be able to simply assume payments. However, the mortgage company may have other opinions and regulations about what they are willing to do.
I am under the impression that a wife's credit is considered separate from her husband's, unlike what it was through about the 1970's, when a woman tended to lose all individual identity of any kind upon marriage. But if you fell off the radar when you got married, it may be more difficult.
Oh, and the other thing about depending on alimony is this: Will he actually pay it regularly? Is he working? What's the chance that he might appeal to the court to reduce his payments? That happened to me, when about a year after our divorce my alimony was cut by two-thirds. I was actually in the process of buying the home I currently live in at the time. Luckily for me I was working and I had other assets that meant I could cover the mortgage, even with that reduced alimony.