Texas, huh? You picked a wonderful time of the year to do your coolant changeout . . .
Anyway, presuming your current coolant isn't clouded with corrosion or precipitates, and transparent except for marker dye coloration, you won't require a chemical flush. As to refilling with new antifreeze and using the Flush-'N-Fill kit, NO to the former unless the prior coolant is the same or very,
very closely related to the new fill. My objection to the Flush-'N-Fill kit is that it relies on tap water which will leave some amount of mineral content in your cooling system depending on the mineral count in the water. Unknown concentration of mineral content opens the door to the possibility of mineral precipitates coming out of solution later on and playing havoc over time with waterpump seals. It's a #$@&ed nuisance timewise (believe me - I know), but doing multiple distilled water flushes (idling upon each refill until the thermostat opens - this happens once the radiator fan comes on
twice and the upper radiator hose is
hot) is the best way to avoid that. Doing the flush kit first, followed up with multiple distilled water flushes doesn't save you any time, and you really won't know when you've achieved stop point 'cause there's no marker dye at all left to guide you. Once the drained effluent is water-white clear or very close to it, you're good to go with the new coolant concentrate*, comfortable in the knowledge that anything previously in the system that's water soluable is essentially
gone for all intents and purposes. (I don't obsess further if there's just a hint of the previous coolant's marker dye color still present.) At about sixty cents a gallon for distilled water at Wal-Mart, the extra expense of four or five gallons of distilled or de-ionized water to cover the flushing and final dilution for an intermediate size car is a negligible expense. When you do refill with your chosen brand of antifreeze/coolant concentrate, use one of the needle-gauge type Prestone or Victor hydrometers to adjust to the final dilute concentration.
*Even if you bother with removing the block coolant drain plug(s), there'll always be some coolant or water left in the passages' nooks and crannies. (I prefer to let sleeping dogs lie, myself - at best it's a gushing "shower" each time you remove the plug(s), at worst, the plug(s) is/are difficult to reach while lying flat on your ..."back side".) If you use pre-mix coolant, it'll be impossible to achieve an actual 50/50 concentration.