I'm gay and I want to live my life with a man and love him. But I know that sex between 2 men is haraam in Islam.
So, can I have a relationship with a guy and all but avoid sex, is that also considered a sin?
This is OK. And this is what many men practice in everyday life with brothers, friends and others.
It is good to love, be it the same sex or the opposite. You are encouraged to develop love for all humans and be merciful.
In Syria, Lebanon and surroundings you will see men walking hand in hand, which Europeans look at with suspicion because they consider that gay. It is a sign of love and trust.
In regard to what is lawful or unlawful the same rules apply to all of us. Thus we should avoid putting ourselves into a situation in which we are tempted to sin, e.g. Khulua, which was originally a concept to avoid having a man and a woman being in a place where no third party is expected to appear (which could lead to sexual intercourse).
In your case I reckon such a rule would apply, since you are sexually attracted to a man. This makes it very difficult to have a close relationship with him. But God knows best and is the most merciful.
In any case, be sincere, honest and try to be good. Don't cheat, (ab)use and respect the feelings and the society you live in. This entails that often it is considered disrespectful to publicly showcase a relationship, depending on where you live.
dictionary.com defines "gay" as: of, relating to, or exhibiting sexual desire or behavior directed toward a person or persons of one's own sex.
if you mean "gay" as defined "by the book" then it's definitely a sin. otherwise if you love a man, like a brother or friend (a brother in Islam is widely used for a close friend or any other fellow Muslim), without the sexual desire this is highly appreciated in Islam. this does not involve marriage, sexual intercourse or even looking at him with sexual desire.
Uh, in all probability it is a haram act as you yourself explained that "I'm gay". Because the mentioned word as you said "gay" can indicate its meaning which is involved of Satan's thoughts and even at many times or even at majority of times it will lead to great sins. Of course as a general rule, thinking about sin is not haram, but it can be the introduction of sin. Also, don't forget that being in that situation as you yourself counted you as a gay with him, will take you to the practical step too (even in less haram acts). On the other hand, we should notice that homosexuality is not permitted form the view of Islam, then being gay is a kind of homosexuality, isn't it?
Hazrat-Masih said to his followers that
"…, but I advice you even don't think about Zina, because…"
Then it shows that thinking about bad acts is not admirable from the view of the prophets. Although I still believe that may be it can be directly a haram act. Because: if looking at non-mahram is haram (by intention of Sin), then generally your looking to your mate(gay) is with the attendance or intention of lust or sin, they it is better for you to repent God and obey Him, then He will give you the best things in this world and insha'allah in the Jannah. Wallaho a'lamo