Is the prohibition on homosexuality only on the act of sodomy or is it for all romantic relationships of this kind?
Committing the act of sodomy is a sin and a crime that is comparable to adultery.
However, just like for Zina, things that lead to this crime are also forbidden. Because Allah says for Zina:
And do not go anywhere near adultery: it is an outrage, and an evil path. (17:32)
The same concept applies (doubly so perhaps) for sodomy.
Furthermore, as Crimson quoted, the Prophet Muhammad (SAW) said:
A man should not see the private parts of another man, and a woman should not see the private parts of another woman, and a man should not lie with another man under one covering, and a woman should not lie with another woman under one covering. (Sahih Muslim)
This clearly forbids any romantic relationships between the same gender. Not even sleeping under the same covering even if they do not commit the crime itself.
So, in fact, there is no doubt that the action is a sin.
However, at the same time, a person is not automatically sinful just because they have desires to commit sins. Everyone has desires to commit sins. And if we do not act on it and choose to ignore the desires, there is no problem:
Allah has forgiven my followers the evil thoughts that occur to their minds, as long as such thoughts are not put into action or uttered. (Sahih Bukhari)
So, even if you have desires for something evil, that does not mean you yourself are evil.
There are more kinds of love than romantic. If you are incapable of feeling romantic love normally, devote yourself to loving Allah. There are many people who are incapable of being married due to whatever reason even if they want to. That does not mean those people are allowed to fall into adultery.
Rather they must control their desires, and be patient for what Allah plans for them even if that means never marrying.