They are both grammatically correct. I think it's more common to say
"I feel tired, I'm going to bed."
"I feel tired. I will go to bed."
The second choice sounds like a robot, and doesn't sound natural.
Sure, when you say "I'm going to bed now", it is in the present tense. It can be a sort of extended present tense though, like you are on your way to bed now, the whole process will take a few minutes, etc.
However, considering other use cases... it might depend on whether you are announcing it to a room full of people, or just thinking to yourself: "I feel tired. I will go to bed in 10 minutes."