There's not going to be much description of this sort of thing in the context of actual combat (there may be some in accounts of judicial duels or later manuals on weapons training and fighting). In general, chroniclers and the sorts of people who wrote history in the premodern era were not concerned with the technicalities of the individual soldier. Even for large battles, the details of how the fight took place were often described in vague, simplistic terms. If individual soldier's actions are described at all, they are usually royalty or important figures and the focus is on their deeds. So instead of "Lord So-and-So dropped his lance and drew his sword," we would be more likely to see "Lord So-and-So captured Duke Such-and-Such and was rewarded greatly by the King for his valiance."

That said, it seems more likely from the fragmentary evidence available that a medieval soldier would not have a problem with dropping his weapon if he felt like a better option was available. Weapons broke, got dropped, or knocked out of people's hands. Soldiers would sometimes throw their weapons if they felt like they could get an advantage out of it. This was the case both with weapons that you might expect to get thrown (throwing axes, spears made for throwing, rocks) and less typical ones (a horseman's lance, swords, maces). Soldiers armed with weapons that were difficult to use in particular circumstances (archers, pikemen) were generally armed with other weapons that they could use to defend themselves in case of their usual weapon becoming useless. English archers do not seem to have hesitated to drop their bows and pick up hand weapons to fight in close combat. While it is tempting to speculate how they might have figured out whose bow was whose after the fact, there is no hard evidence that would help up figure out exactly was the process was. Either there was some kind of system for doing so or they simply considered the tedious process of sorting out abandoned bows to be less of a problem than the prospect of losing the battle.