I've used this advice for 15 years to function effectively in cities.
Plan your eating
An essential strategy in any city is to plan your eating so that at times you will need the bathroom, you will be having a meal or snack. That makes you a paying customer, and you are definitely entitled to use any bathroom they have for customers.
Partner to this strategy is do not gorge yourself. Eat lightly so you have room for another snack at another establishment should the "need" arise. To be clear, the "snack"'s solitary purpose is to provide a statutory transaction for use of the bathroom; it can be one biscuit that you share or simply throw away. Since you have no issue with paying for bathroom access, this shouldn't be a problem.
Plan your drinking
This should go without saying, but err on the side of drinking less when you're out and about. You don't need to pee what you didn't drink. The wrong thing is to make full use of free refills, or get the large because it's twice the drink for 10% more, for instance, so you have to re-train those thrifty habits.
And sync your drinking up with your kids' so you're drinking about the same in proportion. That way you'll know how they feel.
Since dehydration is a concern -- actual, medical dehydration can be trivially tested in 2 seconds. Ask your doctor. So if you are worried about this, learn the test and do it as needed. If you are adequately hydrating your children normally, the limiting of fluids a few hours before and during the city excursion will be inconsequential, unless they get overheated, which you'll know because you'll also be overheated. If in doubt, test for it.
Most city hotels have conference rooms and they are usually renting different rooms to different companies at once. So the areas are public access. Just walk in like you know where you're going and seek out that area. Bathrooms are nearby.
Train your kids to go when able
Break them out of the habit of only thinking about bathrooms when nature calls. Teach them to go when the opportunity arises. This is a valuable life skill that you'll be giving them.