Gollum had his life largely extended because he possessed the One Ring. He appeared to be an old, deformed, and twisted creature.
According to this list, he had the Ring for 478 years. His loss of the Ring and the events of The Lord of the Rings are separated by around 77 years.
Did the Ring continue to extend his life after he lost it?
Yes, the Ring continued to extend Bilbo and Gollum's lives after they stopped bearing it up until the destruction of the Ring.
Gandalf himself alludes to this when talking about Bilbo:
it might take a long while for the influence to wear off. ... he might live on for years, quite happily: just stop as he was when he parted with it.
Since neither Gollum nor Bilbo appear to age significantly after losing the Ring, despite their already extended ages, that's a reasonable assumption.
Bilbo had deteriorated significantly by the time he left Middle Earth, which concurs with this conclusion. The Ring had been destroyed by then so any legacy effect it had on him would have ended. Arwen would have only left Rivendell a few weeks after the Ring was destroyed to travel to Minas Tirith, but when she got there she commented
Do you wonder at that, Ring-bearer? For you know the power of that thing which is now destroyed; and all that was done by that power is now passing away. But your kinsman ... is ancient in years now, according to his kind; and ... he will not again make any long journey save one.
Obviously the destruction of the Ring was already having the effect that Gollum also alluded to when he said
when Precious goes we’ll die, yes, die into the dust.
We never got to witness this because of his encounter with the interior of Mount Doom instead.
just stop as he was when he parted with it - this sounds like a typo. Is it supposed to be "just stopped as he was...", or am I reading it wrong?
No.
More than likely his life simply picked up where it left off before he was exposed to the One Ring. Hobbits are known for a longer lifespan than Humans, so likely after he lost the Ring, his normal lifespan continued until he died at Mount Doom.
To use Occam's Razor: The simplest answer is best. If the Ring were sustaining Gollum, literally providing him with life-force, as soon as it left his possession, he would die. Why? He had served his purpose to move the Ring. Once he started staying still and just sitting with the Ring, it found a way to bring others to it and eventually to get away using Bilbo. If the Ring were sustaining Gollum, that would be both a use of energy that could be better served controlling its new carrier, Bilbo.
There are several instances in the story, the Ring would have been better off severing its link to Gollum, particularly before it fell into Mount Doom. Since it didn't, it's safe to assume, Gollum's activity was purely his own, working from the remnants of his mortal life.
Gollum possessed the ring for nearly 500 years and never let it go so its power was complete over him, Bilbo had it a mere 60 years and life in the Shire meant he didn't spend all his time obsessing over it so the rings power over him was less strong.
As long as the ring existed Gollum was a slave to it so yes the ring continued to exert its influence over him, Bilbo on the other hand was very much a Hobbit whos life had been drawn out long by the ring but without it age caught up with him.
It was very apt that Gollum died with the ring as he probably wouldn't have survived its destruction.
The reason Gollum continued to live despite the Ring being absent from his possession for so many years was because after nearly 500 years of having it, the One Ring had transformed him entirely into something else. He was utterly corrupted.
Consider this: Gollum was in possession of the One Ring far, far, FAR longer than any other being (excluding Sauron). A strong possibility would be that just as the Dark Lord had imbued the Ring with a significant portion of his own life force, that same life-giving force became imbued within Gollum. His split personality was not a mere quirk of the mind, he actually was forced to contend with a living entity that had incorporated itself into his body and mind.
Another point to consider: a number of times, some speak of the 'ring wanting to return to its master; purposefully abandoning Ilsidor; having a will of its own'. Then, it did not expect to be possessed by a hobbit (Bilbo). Maybe Gollum had continued with long life because the ring knew the path back to its master would not occur by the hobbit, meaning, 'IT NEEDED GOLLUM TO ASSIST IN ITS RETURN TO SAURON'. The use of magic is quite strange in many places in LOTR. Needing Gollum seems like a plausible reason for extending magic to his life.
Another thing we all may want to consider is that each character may have had other forces in play sustaining their lives as they had "roles" to play in the scheme of things (like Gandalf being returned to Middle Earth after his fight with the Balrog). Eru Ilúvatar's music works in mysterious ways.
As the ring was in fact sentient and an object of pure evil. Perhaps Gollum was the One Ring's back-up plan; the Ring chose not to age Gollum knowing he would seek it out if it fell into the hands of those seeking to destroy it.
