Palace of the Dead does give out ilvl255 weapons, but I wouldnt really advise you to go down that road - at least not without looking at the other options. PotD does require some griding to get that ilvl255 weapon and while you can combine that with leveling a different class there are some better/faster options to get weapons close-ish to that ilvl range (please note that I'm a little biased because personally I dont like PotD to much...):
First off: I dont know what weapon you're using right now, but you can get an ilvl235 from doing just a few PvP-matches. That should work as a gap-closer should you be really far behind.
Now to some real options:
1) Crafted weapons - its an obvious one. Ilvl250 can be crafted - depending on your financial situations or friends/FC-members being able to help you, this might actually be the easiest option.
2) Shire weapons - ilvl260 (can be upgraded to 270). Requires you to run A12 NM seven times and spending 1.000 scriptures (to be precises: you can do one A10S instead of seven A12NM aswell, at least in theory...) - running VA, Mhach and Dun Scaith will give you the item to upgrade
3) Primal-weapons - ilvl 230 (can be upgraded to 240). The version before Shire. Requires 4 runs of A8NM and 1.000 lore - if you do one run of VA and Mhach you can get the upgrade item
4) EX-Primal-Drops; specially Sophia - it might require you to upgrade your other gear a bit before hand and you should at least get that PvP-weapon, but she is on of the easier HW-primals and drops ilvl255 weapons. 10 runs will give you enough tokens to buy a weapon, should it not have dropped.
5) The Relic - yes, its a grind. But it has been nerfed and the first few stages arent to bad anymore, so its an option to get a half decent weapon "quick", gives you something to work on, opens a unique questline to follow, a unique weaponmodel and most likely a headstart on the SB-relic (if you want to settle for FFXIV you might want to consider that)
...there is a very, very slim chance to get a weapon from diadem, but I wouldnt bet on that - just adding for completion.
You could also just say "Screw this" at this point actually, focus on leveling classes and just get the new gear that SB will bring in a few weeks - I expect all highend gear that we've been working towards and hoarding for the past few months now to be worthless within the first 6-7 levels, so probably the first week of SB (depending on how fast you are...). So personally I wouldnt go to much out of my way to gearhunt now - not with the knowledge that in a few weeks there will be drops from teh first dungeon of the expansion being better/on the same level as the weapon you're grinding for now.
So my advice would be: Get what you need to clear the content you want to clear and leave it at that for the moment. Focus on leveling, levels stay! (In that sense: Yes, you could get the PotD-weapon if it just comes with the process)
Get the PvP-weapon or better the crafted one, if you can. If you're willing to grind, you can consider the relic - as I said... headstart in SB and a unique weapon.
Edit: Yes, the Chaos-Armor (DRK artifact) is now only bought with hunting seals.
And I'd like to repeat my general advice: Dont stress to much about getting your jobs to max-ilvl now. At least I would feel as if I'd wasted my time when SB hits and all of that work becomes invalid within the first 5 levels. Grab as much gear as you can from running Dun Scaith, A9-12NM, tomes... but dont stress it to much - once SB hits there will be even more and better options to gear up.