OP, to give you a couple of table service options without the need for ADRs (at Disney Springs and Animal Kingdom, anyway), I suggest you purchase a Landry's Select Club membership ahead of time. It "costs" around $20, but you get a "welcome credit" for that amount the first time you present it at a Landry's-owned restaurant, so it ends up being free. At WDW, the Landry's-owned restaurants are among the most kid-friendly on property: Rainforest Cafe (at Disney Springs and at Animal Kingdom), T-REX (at Disney Springs), and Yak & Yeti (at Animal Kingdom). Select Club membership includes priority seating as a perk -- that means, you can show up to those restaurants at any time, without a reservation, show them your Landry's Select Club card, and be seated just as promptly as if you did have a reservation.
Otherwise, if you try to get into TS restaurants day-of, you'll find that your selections are very limited, and you may be forced to either do counter service instead, or eat at an "off" time (e.g., dinner at 10pm, lunch at 3pm, etc.) The least likely place you'll find any availability will be character meals, so if that's what you and/or your family really wants, you really need to make a reservation for one.
Are you just afraid that your plans might change and you'll have to pay the no-show fee for canceling less than 24 hours ahead? Never fear! If you decide with less than 24 hours to go that a reservation no longer works with your plans, simply use your MyDisneyExperience app to modify the reservation to some future date, more than 24 hours into the future. Then, a few minutes later, you can cancel it without penalty.
*NOTE: This advice primarily applies if you wanted to dine in one of the Disney parks. Off-site, the sky is the limit. Resorts sometimes have better availability as well, and many of the [privately-owned] Disney Springs restaurants have 2 reservation systems -- one with Disney and one via their own website, so there is usually availability through the outside websites or by calling the restaurant directly, even if MyDisneyExperience/WDW Dining tells you their allotment of reservations is full.