You're not going to find a DPS job that has great healing support. Or even decent healing support. There are DPS jobs that can heal, somewhat, but in general it's really more of an emergency measure and wholly inefficient.
Support for DPS Jobs in FFXIV falls broadly under these categories that a job may or may not offer:

General moar damage type of deals: Out of the DPS Jobs most offer these in one way or another, be it the crit buffs from Dragoons or Monks' physical damage increase or skills that generally increase the damage taken by the target like Ninja's Trick Attack or Bard's Foe's Requiem. Notable outliers here are really only Samurai and Black Mage who are the "selfish" DPS that offer little in the way of group utility.

Resource replenishment: All physical Melee DPS get Goad which restores a single ally's TP over time. All magical DPS get Mana Shift which gives some of their MP to an ally. Physical ranged DPS, that is Bard and Machinist, get skills that gradually increase the TP or MP regeneration of all allies in range.

Resurrection: The one ability you should never have to use that can salvage a run that may otherwise end in a wipe. Sadly aside from the healers resurrection skills are exclusive to Summoner and RezRed Mage so not really relevant to what you're looking for.

If you are looking for the DPS Job with the most varied "support" portfolio I'd point at the Bard. They get skills to replenish the party's TP and MP, Palisade to soften a blow someone is about to take, Warden's Paean as an emergency debuff removal, Nature's Minne (it does have it's uses), Foe's Requiem to temporarily increase the party's damage output, Battle Voice for a Direct Hit buff and the general (if small) crit buff in effect while one of their songs is running.

If you're looking for something that very loosely resembles the Enhancement Shaman class fantasy (at least as I remember it, I quit WoW ages ago) you might also want to have a look at Monk. They use fist weapons, get faster as they build Greased Lightning stacks and get different elemental stances that do things (though generally you'll mostly be using Fists of Fire). For support they only get Goad as a Role Action along with Brotherhood (a physical damage increase for the party) and Mantra (it's like Nature's Minne but as an AoE) but at least it's something.