Avocados are very difficult to root, and generally require at least a few green leaves attached to the cutting in order to succeed, and typically will only root after many months. If the leaves fall off sooner, that means probably no roots will form. The stem will sometimes stay green for many more weeks, but I’ve never had one succeed without leaves kept on it until after new buds begin to swell.
I would normally suggest grafting rather than rooting, it’s a much higher success rate. But either way, I would say prepare the cuttings soon after you collect them, rather than keeping a whole branch and letting it dry out.
Scionwood for grafting should be only hardened stems, any new growth should be removed, and it should have all leaves removed and be soaked in a glass of water for a couple hours and then refrigerated in a plastic bag.
Cuttings for rooting should have at least 3 and up to 6 leaves on them, but the leaves can be trimmed back a bit to minimize water loss and make it easier to cover them with a jar to maintain humidity. Here’s an example of what those look like, though the stem can be a bit longer:
Those can be stored in a glass of water at room temperature for a few weeks, as long as the water is changed regularly. Some will wilt, and should be discarded before they foul up the water. Once they go in soil, they should be covered for at least a few months.