Bfwahahaha! I find shy girls cute. And I'm sure Red does too.
Synchronization is right, Katie. Though, frankly, while a lot of people take me for an "Extrovertive Alpha", I can actually relate to how you feel there -- I also have a bad habit of avoiding eye-contact unless necessary. Being in the shadow of my Islamic mentor, who helped cure my dyslexia and polished my intellectual character, has turned me into an indefinite Introvert. Being reserved is well, but having social connections is also necessary, and hence I try to overcome the restrictions of my nature.
Now getting back to your problem (or something you
think is a problem) -- I hardly think you need to overcome something like that. It is the attribute of the meek and the modest, and is useful for survival purposes (i.e., you won't catch unnecessary attention that you aren't comfortable with). Even in the instance of your grandmother, you avoid eye-contact, subconsciously knowing that you don't want to get into another uncomfortable conflict. So basically, it isn't really as bad as you think.
BUT! A good eye-contact determines character, and exudes confidence. Your confidence isn't determined by how loud you are, but how slow and clear you sound. A firm handshake, a confident look, a respectful smile and a warm voice all open up a friendly social relation even with the most reserved of strangers -- and it can even land you a job! Eye-contact is the attribute of the strong and confident, and of course of the vigilant. Sometimes it even determines a personal association, that you're listening to the person closely, that are focused on them, but it can also mean defiance and enmity. Eyes are like doorways to a person's soul, and they tell you everything you want to know about them (yes, even when they tell you, "It's nothing").
(There are differences between "glancing", "looking", "staring", and "glaring" though, but that's probably for another topic.) So how do you overcome it, Katie? Firstly, ask yourself: Why aren't you confident to look them in the eyes? What makes you so insecure about yourself? Does it relate to some other weakness or regrets you have? Is there something you don't trust about yourself? What are the things that you're still unsure about?
Chances are, when you've become truthful to yourself by honestly answering those questions you can also gain enough esteem to be competent in social situations -- it's all about facing your weaknesses and evaluating them, because honesty is half the battle won. After that, you might need to "train" yourself in extroversion by learning basic acting (in someone's presence, be it Red or your parents, or even other friends); grab some dialogues, learn postures and body language, and try to approach convincingly. Acting is an art and provides you with enough toolset to physically dispense your thoughts into communication clearly and concisely.
And last, but hardly the least, smile when you're amused -- and not just because you have to, but because you
want to. Though, there's one thing introverts usually always possess compared to extroverts: the ultimate Pokerface! It makes you mysterious~