After Constantine, only one emperor, Julian the Apostate (360-363), was not a Christian. Julian made some vague rumblings about disfavoring Christians, but these were perfunctory, and Julian's reign was rather short in any case. These don't really amount to "attempts" to return to paganism.
By the time Constantine (and even Diocletian) managed to recover some sense of integrity, there was little left of Glorious Rome.
Again, you are too broad. Diocletian's reforms crippled the Roman economy, which had already been weakened by a century of almost continual civil war. But Constantine's reign came at the beginning of the fourth century, and at this point in time Rome was still incomparably mighty and in possession of much of her past glory.
Some historians, most notably Gibbon, have argued that Christianity actually hastened or even caused the fall of the Roman Empire, through a variety of factors, including the economic drain of monasticism, eschatological detachment from secular affairs, and cultural rifts caused by theological strife. It is therefore too simple merely to claim that bad morals and a lack of integrity helped the Empire to fall; paradoxically, it appears that the Roman Empire was strongest during its periods of moral decay.
By the time Constantine began his reign, the Roman military could no longer sustain itself and was forced to hire mercenaries to defend its borders. This contributed to a further weakening of an economy crippled by civil war, and forced the Emperor to institute drastic economic reforms that only delayed the inevitable end. With the confederation of many of the barbarian tribes, and given its dwindling resources, Western Rome could no longer protect its borders. The end was assured by the start of the Fourth Century, although the Empire struggled on after that, even managing some semblance of its former glory whenever it could stop bickering long enough to raise an effective army.
While it is simplistic to attribute the fall of Rome to moral decay, it is simply revisionist to pretend that it didn't play a role. Constantine's reign marked the last flare of the Roman candle before the flame was extinguished permanently.