using multiple cables like that is called link aggregation. This requires BOTH devices to be capable of it. It only helps if you have multiple streams trying to cross that path as each stream gets assigned one cable.
depending on the model of switch you have. some have a designated uplink port while most of the newer ones all the ports are the same and it doesn’t matter which one you use.
getting newer hardware isn’t a bad idea. But your issue MIGHT be able to be solved by turning on some QoS settings and prioritizing the video meeting and gaming devices.