Consider ...in one go or ...in one sitting.
at/in one go (British English)
in one single action; all at the same time:
I don’t think I’ll be able to solve all the problems at one go. He ate the whole cake in one go.
See also: go, one
In one go is as natural as I can think of to this native British English-speaker.
In one sitting might be more natural to American English-speakers, but I usually consider this one more-related to eating lots of food, rather than watching lots of TV.
Whole season and Entire season are both fine.
If you want to use in a row, consider changing the sentence so that the subject is plural. All episodes is a good substitute.
- "I watched the whole season in one go."
- "I watched the entire season in one go."
- "I watched all episodes in a row."
- "I watched all episodes, one after the other."