Thank you for your question. I think I have found what I need in an Olympus TG6, and used they are very reasonable. Yes, when printed the photo has to have a sate stamp originating within the camera, added or software originated dates are considered not acceptable by immigration.
I have the earlier TG-5 and like it. But, I'm not sure it will put the date stamp on a picture. It does record date, time and even GPS locations. I don't know how any of those would be on the prints done at a place like Walgreens.