“chodesh Teves is mentioned in Esther.” Because they were in Paras. Everywhere our current month names are mentioned are Post-Golus Bavel. In the PRE-Golus Bavel we have:
EXODUS 13:4 – This day came you out in the month AVIV.
EXODUS 23:15 – You shall keep the Feast of Unleavened Bread; you shall eat unleavened bread seven days, as I commanded you, in the time appointed in the month AVIV; for in it you came out from Egypt; and none shall appear before me empty;
EXODUS 34:18 – The Feast of Unleavened Bread shall you keep. Seven days you shall eat unleavened bread, as I commanded you, in the time of the month AVIV; for in the month AVIV you came out from Egypt.
DEU 16:1 – Observe the month of AVIV, and keep the Passover to the Lord your God; for in the month of b the Lord your God brought you out of Egypt by night.
1 MELACHIM 6:38 – And in the eleventh year, in the month BUL, which is the eighth month, was the house finished in all its parts, and according to all its specifications. And he was seven years in building it.
1 MELACHIM 8:2 – And all the men of Israel assembled themselves to king Solomon at the feast in the month ESANIM, which is the seventh month.
And so forth…