Speaking as a practitioner:
It's not so much the weight (swords are light) it's the size. It can be very easy to trip over your own scabbard if you're not careful, and trying to carry more than one sword is just going to compound the problems as they get in the way.
You'd carry ONE full-length sword on your person — usually either an arming sword (one-handed) or a longsword (two-handed) — as a sidearm and worn at the hip (back scabbards for any sword of appreciable length are NOT practical, and it can be very difficult to draw from that position. That's why in the original Conan movies, Ah-nold MOVES HIS SWORD from his back to his hip whenever Conan expects to need it). Your primary weapon in the West would be a polearm such as a polehammer, halberd, or pike (this is where the katzbalger came in; as a sidearm for pikemen) which would be carried in a "carry arms" position (it always irks me in video games seeing polearms strapped to peoples' backs
). The exception to the "one sword" would be if you also were carrying a doppelhander in place of the polearm, as was often the case for the Landsknechts, and these would be carried in a similar manner to a polearm (those are the very big two-handed swords which filled a roll closer to polearms). Any secondary bladed weapons beyond that would generally be some form of dagger such as the rondel (traditionally worn at the back). And if you're wanting your character to be carrying a large number of weapons, daggers would be the best choice.
If you wanted to carry around multiple swords, you could hang one from the saddle of a horse.
In fact, if your characters' swords have a routine problem of getting stuck in a monster to the extent they can't be withdrawn again...why are they even using swords? It sounds like bludgeoning weapons such as hammers and maces would be a much more practical choice.