The order that works
Generate candidates, check every one before the shortlist meeting, and bring only available names into the room. Nobody argues for a name that cannot be bought.
What to check beyond the .com
The extension your market expects, the social handles, and whether an existing trademark covers the same class of goods. A free domain is not the same as a free name.
Compound names and the alternatives
If the single word is gone, the usual escapes are a prefix like get or try, a compound of two short words, or a different extension that fits the product. Developers read .io and .dev without friction, and a consumer product usually should not experiment.
One thing not to do
Do not buy a misspelling of a taken name in the hope of traffic. It costs the same as a good name, converts worse and invites a dispute you will lose.