I would not play any of the melee classes if you're hindered in one hand. The ability to move freely while executing your abilities is extremely important to melee, moreso than other DPS classes, because they have to be dodging and moving a lot more than ranged classes. You may be better with Black Mage - they have very little to try to do when simultaneously moving, which would make it comparatively easy to control - you're either attacking or moving there. That said, if you're insistant...
Dragoon as a melee is the 'slowest' of the melee classes due to a lack of innate haste, and uses prebuilt combos, leaving them theoretically "easy" to play - but like any class, playing them well is anything but "easy". Dragoon focuses on properly selecting and executing lengthy, pre-built combos to juggle both debuff/buff effects and raw damage. It also weaves more off-GCD abilities between attacks than other classes do - including the powerful "Jumps", which despite the sound of them don't prevent you from taking damage while they're flying around. Despite a higher physical durability than any other non-tank class, Dragoons that misuse their abilities will find themselves exploded due to animation lock. Positional requirements are fairly light until post-50, where one particular attack can lose huge amounts of potency when performed wrong. Dragoons have great burst and capability to amplify the rest of the party's damage.
Ninjas are a DPS class that effectively juggles two rotations - their standard attack rotation, and their Ninjutsu rotation. Their standard attacks are simple preformed combos that are easy to learn the patterns and timings for, making that part of the job relatively easy among the three; they're also the least positional-reliant class and suffer the least from it. The complexity of the class comes from their Mudras - an ability available every 20 seconds, where you manually input a sequence of three skills to 'cast' a variety of skills, from the constantly useful (A personal haste) to more situational (AoE damage field); a lot of the difficulty comes from memorizing and accurately executing these Mudras. They're very capable overall as a class.
Monk is a heavily positional-focused melee with a completely 'fluid' combo system. Unlike the prestructured combos of every other melee class (including tanks), Monks rotate between three different stances with different attacks available as they strike, aiming to maintain a balance of debuffs/buffs/DoTs and raw damage attacks that can swap on every hit. As they complete combos, they also build a stacking buff that accelerates their damage even further; letting it drop off costs you hugely. Almost all of their attacks also benefit from being in the right location, to the tune of 25-50% of the potential damage of an attack, so it's incredibly important to learn. It's theoretically simple, but very execution heavy; Monks might not bring damage buffs to the rest of their team (Although they do provide constant outgoing damage reduction), but their single target potential DPS is unmatched.