Maybe (a retroactively cast) Magic Jar and Glyph of Warding?
The Magic Jar spell (PHB, p. 257) enables a creature to send its soul into a prepared "gem, crystal, reliquary, or some
other ornamental container worth at least 500 gp": there's nothing in the text indicating it couldn't be a magical ring. While in that object, their soul can subsequently attempt to possess a humanoid who is within 100 feet of the container.
The spell has quite a few details, and you should study it to make sure you understand all its ins and outs (for example, if the container is destroyed, even while the soul is possessing a new body as a feature of this spell, the caster [i.e. dragon] will die). But the most problematic part of the spell's description for your particular case is the fact that it has a 1 minute casting time.
If this dragon was killed in combat, it wouldn't have had time to concentrate on this spell for a full minute in order to cast it. But it might have been able to set up a Glyph of Warding (PHB, p. 245-246) a long time ago to prime the Magic Jar spell with a Spell Glyph. This could have served as a last ditch "escape plan" when it knew it was definitely going to die. Glyph of Warding's spells activate without having to undergo a longer casting time, so the spell could be set off in the midst of combat.
The only problem about this plan is that when you cast Magic Jar, your body becomes catatonic. The dragon would have needed to activate the spell glyph (by doing some pre-arranged trigger for the glyph, possibly something that costs a Reaction, like casting the Shield spell) just a few moments before it died, so that no one in the party would realize this was happening. If you have a party with members who have very high Passive Perception bonuses, it may strain believability that no one noticed the dragon fall down insensitive before the final blow was actually struck. But if this seems believable to you, it's a viable way that the dragon could cheat death (for the moment anyway: again, read the Magic Jar spell carefully to understand its limitations).
But how did the party Identify the ring?
There's one more important detail you need to think about when considering this strategy: how did the party realize that the ring is a Ring of Mind Shielding? If they took a Short Rest to study the item to learn its properties (DMG, p. 136) you should be fine: they would learn what uses the item has as a magic ring, but nothing else. However, if they cast the Identify spell on the item you could have some trouble because when you cast that spell on an object (PHB, p. 252):
You learn whether any spells are affecting the item and what they are.
Even Nystul's Magic Aura may not help you hide the fact that the Magic Jar spell is affecting this ring, since that spell only lets you choose the school of magic that appears to be affecting an item, which may not overrule an effect that tells you the specific spell "affecting the item." And while the ring is technically not the target of Magic Jar, it is certainly "affecting it" by causing a dragon's soul to reside within it. The dragon could cast Nondetection on the ring to make the Identify spell unhelpful, but this only lasts 8 hours (and it's a bit unrealistic to think the dragon regularly spent a 3rd level spell slot casting this spell on the ring, just in case).
If you simply told the party that the ring was a Ring of Mind Shielding without going into details about how they discovered this fact, then you should be in the clear. But if the party members specifically mentioned casting the Identify spell on the ring, then you may need to reveal the Magic Jar spell to them, which will likely make the dragon's stay in this dimension quite brief.
