The AP 'best for swordsman' thing is a myth.
AP has high mobility, large clip and fast firing rate, but each bullet is fairly low damage by itself and you need to keep up a high firing rate in order to do a lot of damage. Conversely, the Drivers have mediocre mobility, a clip of two and average firing rate, but the bullets can ricochet up to two times and the charge splits into four separate bullets that can bounce another two times. This makes it readily able to take out turrets in a single shot, or rapidly eliminate mobs from a distance. (AP is also quite expensive to get, and the charge isn't especially useful in most situations.) It would really depend on your playstyle.

Snarbolax is a Shadow Lair armour. It is therefore quite difficult to get, especially if you don't already have an appropriate armour for running the Lair. Vog is therefore probably a better choice to start off with.
Your swords look fine. Your bombs also look fine.

If you're into swords, then you will eventually want to have three swords (piercing, elemental, and shadow), but you never need to carry all of them at once. Consider just equipping the two that are most relevant to the level, along with one gun (often elemental, but not always), and one bomb.

You mentioned that you're trying to do all of this cheaply, without crafting multiple specialized sets. Consider maintaining just three weapon slots. Start working on an elemental gun, piercing sword, and shadow sword. And maybe a bomb, if you like. Bombs are fun. Then add more weapon options, including an elemental sword, later.

Aureate has explained why the alchemers outperform Argent pretty well. When people say that Argent is a good sidearm for sworders, what they really mean is that Argent is easy to use. They're assuming that a sworder isn't interested in really learning his gun very well. If that describes you, then go for Argent; it's not bad. Or go for Polaris. But go for alchemers if you are willing to make an effort to use guns well.

As my sword guide and Aureate mention, Snarbolax armor is hard to get. It can be crafted only in the Shadow Lairs. Surviving a Shadow Lair is not easy, even if you have 5-star gear and a decent party. Alternatively, you can buy the armor unbound from someone, but that is insanely expensive and not gratifying. This is why you should focus on Vog Cub and Skolver (and maybe Chaos) for your first sworder armor.

If you have the Snarbolax set, then just roll with it. The shadow defense is more valuable than Skolver's piercing defense. The only issue is that you won't have any fire or elemental protection for Firestorm Citadel. One piece of Vog Cub might help you out there. But try it with Snarbolax.

For DPS bombs, the most popular choices are Dark Briar Barrage (piercing, no knockback), Nitronome (like a normal version of DBB, with high knockback), and Dark Retribution (shadow, no knockback). Ash of Agni delivers nice DPS against crowds. Voltaic Tempest delivers nice DPS against tight crowds, because shock damages nearby monsters.

Yes, I post a lot. I like this game. No, I don't work for Three Rings. Cheers.