By the time of the Mongol invasion, the Abbasid lands where already somewhat divided between squabbling governors, basically like the robber Barons of France and the King being secluded to Paris, but this time the Caliph to Baghdad. However, the Muslim lands still held immense wealth, libraries (particularly Baghdad) and so on. It was still a functioning society. The Mongol invasion and razing of Baghdad and the Middle East did two things:

It massacred Baghdad population, one of the largest if not the largest in the Middle East, it destroyed Baghdad famous libraries and infamously the tiger Tigris reportedly ran black with ink as the Mongols threw the books into the river, lost forever.

But it was not only these famous events that served to end the dominance of Syria and Iraq in the Muslim World, but the population change caused by the Mongol invasions which emptied the region and pushed population westward into Hejaz, Egypt and western Anatolia. This ensured the stagnation of the region, which remained a wartorn wasteland for decades to come.

The population never recovered. Damascus and Baghdad were destroyed utterly by the Mongols and even when recovered by the Muslims eventually, they never returned to their status. Muslim power and wealth remained in Cairo, Cordoba and later on Costantinople.

Fast forwards from the 1200s to 1500s when the Ottomans took Syria and Egypt from the Mamluks and the Abbasid Caliph (under their protection in Cairo), the Ottomans never really tried to revitalise Syria and Iraq. I read once, but can't remember the source, that the population of Syria and Iraq after being razed by the Mongols did not recover to pre-invasion levels till the 1900s. So you can just imagine 700 years of devastation, social stagnation, no progress.

So yes, while there was a lot of disunity in the Muslim world at the time, the power and wealth of the Muslims lying in historical centres of Baghdad and Damascus still flowed and was prevalent, until they were destroyed and displaced. The Mongol invasions really changed the history of the region, and most likely the world.

To give you a proper answer: I don't like the contextualization that "Golden Ages" supposedly gives a reader, because it effectively means nothing, and does not help us understanding anything. Not to mention is gives a very Arab-centric understanding of Islam, when in fact Islam now spanned from Iberia to West/East Africa to East Asia.

So did the Mongol invasions of the 1210s, culminating with the sack of Baghdad in 1258 and the sack of Damascus in 1260 mean the end or a "decline" of Islam, or its expansion? Absolutly not, in fact the reverse is true.

The arrival of the Mongols, much like the [Oghuz] Turks 200 years before, was definitely a watershed moment for the Middle East (not all of Islam). However it was the Mongols, under the Ilkhans, Chagatai and the Kipchak Khanate (Golden Horde), which helped to spread Islam, and led to a closer cross cultural exchange with [Yuan] China. Not to mention the many great works which Mongol patrons committed to, such as the great architecture of Sultaniyya or the Shahnameh, not to mention the continuous scientific work, and innovations in government administration.

During the period post-1300, according to Dr. Richard Bulliet, Islam spread the fastest outside of the Middle East, and most of the world's Muslims today, come from converts, post-1300.

So to say that Islam was in decline, is utterly false. For sure Baghdad went into decline, and power moved elsewhere; this had its influence limited to the Arab world, specifically the Levant and Iraq. Power bases in Delhi (the eponymous Sultanate), Cairo, Aceh, continued to grow and expand in their own right, without any interference from the Mongols.

Sources:

New Cambridge History of Islam (2010)

Richard Bulliet lecture series (Columbia): History of Iran