The Kingdom of Jerusalem was always prone to dynastic conflict and backbiting. On the death of Baldwin II, Melisende was overshadowed by her husband Fulk, who proceeded to take over the Kingdom and annoy most of the Latin Easterners who resented his presence and authority, despite Melisende being the heir (and a much more competent ruler in her own right).
What if she was able to take power peacefully, rather than defeating Fulk in a palace coup a few years later?
The only way for Mélisande to inherit alone the throne would have been to remain unmarried, and frankly, it would have been quite unlikely to succeed.
Indeed, the military function of the yerosolemite kingship was particularily important, compared to other latin variations, and the sheer power of Latin nobles within the kingdom would really prevent going around, especially with Zengi pressuring Latin states since one decade.
The precedent set by Alix of Jerusalem when she tried to rule alone Antioche, one year earlier, was clearly onto everybody's minds : trying to by pass the paternal royal's authority, she went as far as promising to acknowledge Zengi as her suzrain. Even if you would likely get nobiliar support, as Alix did, I don't think it would be enough to prevent a conflict.
Now, could have a different husband would lead to a different working relationship with Mélisande? As Louis VI was asked for his opinion about who would make a suitable successor, he didn't have much choice searching around : Foulques was the only one both skilled enough and without a spouse at this moment (that, and getting rid of a too skilled vassal might have been another "good" point on his favour).
Maybe that with a Conan of Brittany that would be unmarried for a bit more...But really, Foulques seem an obvious choice with a 1130 PoD.
That said , Foulques wasn't that incompetent as you made it.
Let's be clear : there wasn't a king that yerosolemite nobility, especially the largely autonomous rulers of northern Latin states didn't complained about at some point, mostly because the yerosolemite kings often tried to enforce their authority, and it did clashed big time with the more autoritarian style of Foulques, that he demonstrated as Count of Anjou and Maine. But giving Latin nobles the free reign they wanted would have been an actual demonstration of utter incompetence, when you see to which extend it failed later on.
Foulques arguably did his best to secure the kingdoms' holdings trough a policy of fortifications, which helped to actually defend relatively well the territory, and Hugues de Jaffa's rebeliousness did a lot to soil Folques' reputation when his main concern was to curb down a too powerful and influential vassal.
On this part, I don't think we could say Mélisande was "overshadowed" by her husband : at the contrary, she did had a lasting influence on politics, maybe as much as preventing Hugues' fuck-up to turn on an outright civil war : how she dealt on the political matters after Hugues' exile show pretty much that she could and did use a lot of political leverage : Foulques had power only as much that he maintained a working relationship with Mélisande.
It doesn't mean he didn't tried to enforce his sole power on Jerusalem, and actually attempted to be acknowledged as sole king early on : but Mélisande basically bitch-slapped and put him in his place.
The 1136 events on this regard cannot be really seen as a coup (even if it's a major strong-arming), as it would imply she regained back her power : she more or less did kept in in face of Foulques attempt, and prooved him at the first opportunity.
but I just remembered that Jerusalem had a leprosy problem.
Not that much : it's just that while it might have been more present in Near East than in Europe, leper was the object of some panic trends. It could be compared to AIDS in matter of how it was seen (and not what it was) : a dangerous and transmissible disease.
Considering Jerusalem's genes as naturally predisposing to leprosy might be a cosmical exageration on this regard. I don't even see how it's related to Foulques in any way.
Is it possible that with a few years of better rule in Jerusalem, the fall of Edessa could have been prevented?
You'd then need Foulques to stay alive. Under his reign, the yerosolemite kingdom reached its greatest extent, and he did have a military mind sound enough to fortify at the right moment, when Zengi was busy elsewhere.
Manassès de Hierges, that was more or less in charge of military matters, was simply not fit for this role (as his haughtiness and leadership during the second Crusade can point)especially not when the northern princes of the kingdom were either incompetent or too rebellious to give a hand.
Foulques was more or less of a jerk at times, but it was an actually competent more or less jerk at times.
What kind of marriage would be arranged for Melisende so as to prevent a similar situation to OTL, in which Jerusalem was wracked with internal conflict in the 1130s rather than focusing on, say, taking Ascalon once and for all, or preventing the rise of Zengi?
With how the Kingdom of Jerusalem was made, as in an idealized feudal kingdom, where big nobles weighted a lot on internal matters and where too important and maintained demonstration of authority were considered abusive, the kingdom was bound to be wracked with internal conflict : every latin kingdom in Europe was, and Jerusalem was at least going to have the same and more likely even worse.