If your cooling system is working fine, then DO NOT FLUSH. Flushing often does more harm than good, especially if you use a chemical flush.
You can get this coolant at your Mazda dealer. You can also look into using Peak Global or Zerex Asian. Look at the section on the label that says “Meets the following specs…” and see if FL22 is listed.
FL22 is a longlife ethylene glycol antifreeze. The basic antifreeze is common to most brands. Most brands use a Hybrid Organic Acid Technology (HOAT) for keeping the cooling system clean and corrosion free. Different auto manufacturers have made slight changes in the HOAT formula and recommend it for their vehicles, but since you are not under the manufacturers warranty, you are not obligated to use this formula.
Any long life, mixes with any color, universal antifreeze will work just fine. The FL22 from the dealer is already premixed at 50/50 with deionized water. You can use any 50/50 premix or mix your own with a full strength antifreeze and distilled water.
You don’t need to flush if you don’t have a problem. You are mainly replacing the corrosion inhibitors so a little of the old coolant left behind will not hurt anything. It will be less harmful than some of the contaminants that often get into the cooling system from flushing.
Edit: If using the FL22 will make you sleep better at night, then use it. It probably isn’t that much more expensive than the brands available at your parts store or big box store, maybe a buck or two. It may be just a hassle or a long drive for you to get to a Mazda dealer to get the stuff. You might try Amazon too.
Edit: Mazda has two coolants, both HOAT long life coolant but one is not FL22. Amazon does carry it. IMO, both would work for you as would any other long life HOAT coolant, but if it makes you comfortable, use the FL22.