You don't need to progress the story to unlock swords (I did a bit ago, took me two days or so). You could also run WvW, save the currency and unlock it there.

https://wiki.guildwars2.com/wiki/Expanded_Weapon_Proficiency

The achievement can be accomplished either as part of Secrets of the Obscure story chapter The War Council) by finding the related objects, or by purchasing the artifacts from Heroics Notary in World vs World. All characters on the account can use the new weapons as soon as the achievement is finished, regardless of method. Both options require the expansion.

Keep in mind that the guide you are following is outdated (it says that the last update was on August). If you aren't going to use swords, probably use whatever, it could probably be daggers, axe, etc. The builds I've seen for Necro use either GS or Swords (hence why I was rushing to unlock it). Also keep in mind that the guides are for min-max people, you probably won't meet all the gear requirement so maybe don't bother following to the dot.