Just a few observations to make, which may or may not mean anything, as I’m not an expert of any of the sort.

There are several reasons to maintain eye contact in a conversation, & respect is only one of them. Never removing your eyes from someone is also a display of authority, quite literally “I’m watching you.” It’s used with silence as a power play in business negotiations with the one who finally speaks first as being the weaker or more subservient of the two, as the counteroffer is where people start getting shredded. Staring is used by predators to size up their prey before attacking. Staring, again with silence, is a tactic used by law enforcement to make a potential witness crack & spill the info they have. Staring at someone can motion to them that you expect them to do something, as well.

Pay attention to your eyes. Are they relaxed? Do you appear expectant, & the other person doesn’t know precisely what you want, so they babble & fidget? Are your eyes widened, as that’s how predators analyze angles of attack? Do you appear judgmental?

Pay attention to your body language. If you’re staring/glaring, are you leaning towards them, looming & poised for potential attack? What’s your tone of voice? How sharp is it? Do you project too much & need to use your “inside voice”? How emphatic are your head movements? Are they smooth & liquid, or are they jerky & unpredictable? What are your arms doing? Are they crossed, thus “blocking” willingness to receive reciprocation? One in a pocket, one by your side, in a relaxed pose? If you move them, are the motions fluid, or are you flailing around like a literal muppet while keeping eye contact, leaning forward, & projecting your voice?

She may be looking around because something about you makes her nervous. Or she may have been raised in a home (or may be in a current situation) where prolonged eye contact ended/ends in negative consequences, so sustained eye contact is uncomfortable. Without analyzing yourself then flat out asking her, it can be difficult to know precisely why a specific individual reacts the way they do.