As Frank mentioned, you can deposit them at the bank where you have your account like any other check. Simply make them out to yourself, counter sign, and endorse. This will be faster, free of any fees (unless your bank charges for deposits), and you should have almost immediate access to the funds. The only restriction, in the US, is that they are USD checks, not anything in foreign currency. Also, remote deposit of traveler cheques using a cell phone app is not supported by most banks, they need to have the cheque. The AmEx site is the best route for foreign currency checks as their fees for exchanging the cheques will be a lot less than what your bank will likely charge if they will even accept them.
I can't stress enough that you should cash any traveler cheques you have in your possession, no matter the brand, as soon as you can. They may still be sold by AmEx and retain their value forever, but they are useless when traveling outside the US and nearly so domestically. No one knows what they are anymore. Merchants don't want to take any kind of check, especially from anyone who is not a local resident, so will most likely refuse them. Banks where you have no account don't want to deal with you or them due to the large number of counterfeit ones which means the bank no way to recover the funds if the cheques are bad. If you try to spend the cheque in a foreign country and it is not in their currency, merchants will refuse them and bank won't even talk to you if you don't have an account with them.
Traveler cheques served a purpose for a very long time. You could carry whatever amount of funds with you where ever you went with no fear of loss because they were replaceable immediately at any AmEx office (if they were Amex). Unfortunately (or fortunately depending on your view), with the presence of ATMs on nearly every street corner and debit cards that work around the world, their time is gone. Their only use going forward should be as a display in a travel museum about how things were last century.