It's really hard to look at both eyes at the same time, and moving from one to the other feels weird, because it doesn't look as if that's what other people do.
Plus, if you do manage to focus properly you have to really concentrate on staying there, and that feels like proper staring (which probably doesn't look nice) and it distracts me from what's being said ...so then my eyes gravitate to their mouth. Not that i'm a lip reader, but it does seem to help, especially if there are lots of other noises around us.
I also feel myself staring when i see a beautiful flower, or e.g. an attractive face ~ flowers are usually russet or burgundy ish with a soft but not too velvety texture ...they kind of swallow me up but i don't mind coz it makes me feel warm and happy inside; the nice faces are usually very relaxed and gentle but with an alertness in the eyes that seems to see past my eyes, to who i am. Of course, god only knows what's really going on inside their heads, especially if they catch me being 'mesmerized' (which i stop myself doing as soon as i notice that's what i'm doing coz i don't want them to think i'm falling in love or something ~ which is probably one of the reasons i usually register people's presence with my peripheral vision).
When i have fallen in love and my 'stares' have been warmly received, i can hold eye contact a little longer with that person, but if they don't say anything i soon get uncomfy and have to turn away.
I'm really only able to maintain some semblance of normal eye contact when someone's made me angry, though in truth i'm taking notice of their eyes, mouth, eyebrows, cheeks ...anything that might help me notice a change in their mood ...so i can calm down as soon as they do ~ tho if they smile at me when i'm in that state it's likely to make me worse coz it doesn't make sense and i don't need them making me confused as well!
When i'm enthusing, i don't look at people coz my mind is usually picturing the thing/s i'm talking about and i'm sort of looking inwards ~ thankfully that doesn't seem to be regarded as rude.