It may have been said already....
I don't think of "feeling" present - I think it's a problematic thing in the English language, that we easily use "feel" as a generic way of describing all states of being.
As in "I feel like they're criticizing me" instead of "I think they're criticizing me"
It's the thing about how thoughts affect emotions, and how easily we can conflate the two. You can think you are being criticized, and that then may cause you to feel ashamed, or sad (or anything else really) - and that feeling of shame can quickly trigger a secondary emotion, like anger or confusion, which then causes you to act defensively.
That's just one potential set of events. But the chaining together of thought --> emotion --> secondary emotion --> action/behavior is something that we easily generalize into "I feel [whatever]". And that can keep us stuck in a loop of emotional reactions, where we get overwhelmed and basically forget how to think.
To me - it's "being" present. As in, I am present, or I'm not. When I'm present, I'm aware of - literally - the present moment. I can see (and hear, smell, touch, recognize) my environment and the situation at hand...and along with that, I am allowing that present moment to fully occupy my attention - or at least, as fully as I'm able to be aware of (since there's always things running in the background that we are processing without conscious awareness.)
If I try and sort out how I feel about being present? Well - to me, the whole point of being present is that it's a non-judgmental state of being. I could be present and also feeling sad - but if I was truly in a non-judgmental state, I could observe the sadness without having it trigger a chain of other emotions, thoughts and behaviors.
That dovetails into some mindfulness theory, and I am far from being an expert in mindfulness.
But it's helped me a lot to get out of the habit of describing everything as "I feel".
And I definitely have talked with my therapist about it - he's pretty good at not defaulting to "feel" for everything, but he'll sometimes ask me if I "feel" present, and I just remind myself that he's asking about my state of being/awareness, not my emotions.
I think of the opposite of "present" as "distracted" or "dissociated" (those are obviously two different things)
And the opposite of "grounded" is "flooded".