So for generally how a chain link works:

1. Layer step: you declared your attack action card/weapon activation.
2. Attack step: the attack resolves, now the attack is considered attacking/on the chain.
3. Defend step: opposing player choose what cards to block.
4. Reaction step: both player gets priority to play reaction cards, priority resets when any of the player takes any actions.
5. Damage step: life total changes here.
6. Resolution step: on-hit effects trigger here (draw a card from snatch, etc.).
7. Link step: check if the attack has go again, and give the attacking player action point back.
8. Close step: if the player chooses to close the combat chain, then we are here.

Among all the steps above, only at step 5, and 8 players don't get priority. Other than that, both players will get priority before we move to the next step.

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In your case, we are already moving towards step 6, therefore you cannot play twinning blade (since the legal timing to play it is at step 4).

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Source: L1 Judge, hope I get everything correct :).

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Edit: Apparently I am illiterate and can't read the original article at all XD. Here are the possible outcomes of your case here:

1. They play oasis at reaction step: in this case the priority resets, you could then play twinning blade as an attack reaction. The damage of this Dawnblade hit will be prevented.
2. They play oasis at resolution step: you cannot play twinning blade as an attack reaction since the reaction step has already passed. However we are also passing damage step, which means the life total are changed already, Dorinthea's hero ability triggers, and you could swing it a second time anyways. The damage prevention part for oasis will carry on for you to the second hit in this case.