If I started a new game as a woman would I be able to marry another woman? If not, is there a way (possibly using the console) to have a same-sex marriage?
The Unofficial Elder Scrolls Pages States that:
As long as you want to marry an NPC that wants to be married, regardless of race or gender, you can marry them. So, interracial marrying is also possible.
Yes it is possible. Additionally, if you see an NPC you like but can't marry him/her, you can use a console command that changes their status.
I think this is the command:
Note: Be very careful when using this command. If you use it on an important NPC it can break any quests related to that character.