A functioning alcoholic is someone who drinks excessively, experiences adverse effects (such as the feeling sick that you mentioned), but still manages to maintain friendships/relationships and get up and go to university/work each day. That state of functioning deteriorates over time.
Going to a GP about this stuff is scary, I get it, I've been there. You feel like you're taking time away from other patients. What I personally did was look into alcohol and drug abuse centres in my local area* first. They're incredibly helpful - and if you try and say "I'm not sure I should be here" or downplay your right to access to healthcare they will remind you that everyone deserves to be healthy. I eventually did later go and see my GP and ask for liver function tests at my partner's request, and the excessive drinking (3-4 drinks daily) thankfully hadn't done worrisome damage but it absolutely could have if I had kept going at the rate I was. My GP also offered to arrange counselling but I preferred to stick with the centre I had sourced myself - it gave me a greater sense of control over the situation.
Seeking help for alcohol abuse isn't something shameful. Many people seek help in order to quit or reduce how much they are drinking. It also doesn't necessarily mean you're committing to giving it up forever, if that fear of never having it again is what is holding you back right now (this was a big hurdle for me personally!). I managed to get my drinking down to healthier levels (<14 units per week) with support from a centre and my family & friends holding me accountable. Not everyone is able to start drinking again, but you've a better chance of being able to do so if you do something about it now rather than allowing yourself to fall too far.
*If anyone reading this, even years from the time of posting, wants recommendations for LGBT+ friendly substance abuse centres in Glasgow then you are more than welcome to contact me.