Is "I am running" or "I run" grammatical to say while running?

when you say a word, you do the action the word describes

This is correct but insufficient: when suggests that the only requirement is that the saying and the action occur at the same time, irrespective of causation; but that does not describe performative verbs accurately.

If I say, "I apologise", by saying "apologise", I make my apology.

The word by is essential: it tells us that saying "I apologise" is the cause of the action. Or, better: saying it is the action; saying it constitutes the action.

When you say, I run, your utterance alone does not make you run. Running and saying I run are not the same action (saying I run is an action of saying, whereas running a marathon is not an action of saying). That is why run is no performative verb in this context.

If you can say I run without actually running, then it is no performative verb. This is the 'negative' test, which is quite reliable. You can't say I promise without actually making a promise, which is why promise is a performative verb: the utterance and the action cannot be separated, because they are one.*

*) You can use verbs that are normally performative in a weird way, such that they are no longer performative; but that is exceptional.