Yes, Judaism did not cease to seek after converts until they were banned (by the Romans, in the fifth century CE) from doing so. But on the whole, in the Roman Empire at least, they were always pretty low-key about it.
A lot of the data about people being ‘god-fearers’ from the 1st century is confused; some of those accused of ‘Judaising’ were probably actually Christians.
Quoth DaphneBlack:
If you mean worship of the Greek pantheon, Zeus, Hera, and the like, that was already dying out as a religion. By the time of Jesus, most Romans regarded the Greek gods in much the same way we might regard Superman, or Uncle Sam: Fun to tell stories about, and symbolic of concepts or ideals, but not really beings to be worshiped in their own right. And the Roman religion, worship of Jupiter, Juno, and the like, had been distorted into unrecognizability long before that, from Greek influence. This is part of why the “mystery cults”, including Christianity, were able to get such a strong foothold: They were pretty much the only option available for religious spirituality.
The older Roman pantheism had indeed almost entirely becoming conceptual by the time of Christianity, to the point where they started adding deities which were explicitly nothing but concepts. (That one always made me laugh. Those Romans were cool!)
Robert Silverberg’s Roma Eterna is set in a world where neither Christianity nor Judaism ever exist, with the result that in the (equivalent of the) 20th Century the Roman Empire survives and rules (or predominates) all the world.
The model that this was ‘dying out’ assumes that it ever was. I put ‘paganism’ in quotes precisely because most people think of it as the worship of the Greek gods, or the Greek gods with Roman names. This was a (relatively small) part of the Roman ‘state religion’ before Christianity, but was by no means the largest part of it. The worship of the ‘household gods’, the Lares and Penates, continued very strongly (in many cases concurrently with Christianity, in some form) and the ‘imperial cult’ – a set of diverse practices related to worship of the emperor(s), living and dead, was going very strong all throughout the pre-Christian imperial period. The ‘representative’ gods did proliferate in this time period, but it was not as if Romans suddenly decided to stop worshipping one set of gods and start worshipping another, more abstract, set; Roman ‘paganism’ was not one system.
The one thing that held ‘Roman’ religion together were a series of rituals, involving rites in temples, and some kinds of animal sacrifice on appointed days, as well as various festivals (fully a third of the year was ‘nefas’ (whence ‘nefarious’) meaning no business could be conducted on these days for religious reasons. This was not, indeed, a deeply ‘spiritual’ set of beliefs, but I don’t think it is accurate to say that ‘Roman religion’ was dying out; it was changing, becoming perhaps more ‘secular’ to modern observers, but it is not right to believe that Romans did not care about their religious practices. Why Christianity in particular was thought to be threatening was because Christians refused to participate in these rituals – rituals believed to preserve the empire!
Unlike other ‘mystery cults’, Christianity demanded a withdrawal from Roman society as such, as expressed in these rituals (and other civic responsibilities).
For a good example of how second-century Romans thought about such things, a good read is Celsus’ ‘On The True Doctrine’ (trans. by Joseph Hoffman, pretty badly, but it’ll do). Celsus wrote in Greek and lived around 180 CE; he was of a philosophical bent but familiar with Judaism and various flavours of Christianity. He articulates how educated Romans of his day thought about ‘religion’ quite well.
Sorry for the long post, I can only blame grad student’s disease.
(How) does it deal with the fact that the pre-Christian 3rd century empire had severe problems with revolts at the frontiers and economic disasters?
IIRC, by judiciously applied handwavium.