Spoilers for Avatar: The Legend of Korra Season 1-3. below.
Korra became able to enter the Avatar state at the finale of Book 1, and we have seen her using it numerous times in Book 2 - even for cheating in a game with Tenzin's children. Even after the happenings of the series's finale, she is clearly capable to go in and out at will.
Still, she never tends to use it when she's in actual danger.
Not for a moment while she's fighting with Zaheer and co., though she and her friends are definitely in great danger. She even loses, without ever using it.
Later she and even her Avatar self is in much more direct and serious danger: being chained up, the Red Lotus want to poison her, and kill her in the avatar state, ending the cycle for good.
She just shakes her chains desperately. I find this really surprising, especially since, as demonstrated under the effect of the poison, she is clearly more than capable of escaping and defeating all the three remaining RL members in the avatar state. It seems had she do this sooner, she could have put an end to the fight much earlier, and never would've got poisoned. And (S04 spoiler)
before season 4, she never had any reason to fear going in the Avatar state. She had hallucinations only because of her poisoned body and fearful mind.
Right, the Avatar state is described as a tool for self-defense by Avatar Roku in TLA, but if avoiding being poisoned isn't self-defence, nothing is. It's also risky for the possibility of the cycle being broken, right, but that's what Zaheer was planning to do anyway. And various avatars used it many times for defending others (see e.g. Avatar Kyoshi separating an island from the continent, or Avatar Roku fighting a volcano to buy fleeing people time). Korra itself used it for things from energy-bending and soul-purifying (see S03E01) to cheating - why not for "really serious business"?
Why is Korra so reluctant to use the Avatar state for fighting?