Hey Elysha,
I hardly ever respond to the comments posted now, but I did more frequently when I lived in Goteborg for a year 2011-12. I totally relate to what you are experiencing, it was the exact experience I had there as well.
I am from (and now live in) Vancouver, Canada. I moved to Sweden with my Swedish boyfriend (still together) with this grand idea in my head that I would work hard, find a great job, learn Swedish... all would be well, just as the Swedes had promised. They said "everyone speaks English, you'll get a good job no problem." Well, turns out it's incredibly difficult to a) job hunt in Sweden, and b) Swedish is very much a requirement.
Before this gets too depressing, I will say that we have only moved back to Canada temporarily, with the plan of moving back to Sweden in a few years once I have more schooling (Master of Architecture). However, instead of Goteborg, which didn't serve us too well, we are looking at Stockholm as most international companies in Sweden have their offices in Stockholm, meaning perhaps most English-speaking jobs.
A couple things I did wrong when I was there, that I have since learned from:
1. I didn't stick to my SFI course - I hated it. I am a person who loved school, so that's saying a lot. I couldn't stand the 30+ students all at different learning levels with little to no attention from the teacher, and the classes were so boring. I wasn't stimulated at all. Anyways, I wish I had stuck it out, or found a way to continue with my courses like more seriously looking into the folkuniversitet courses. They are costly, but I think a much better experience.
2. I didn't seriously enough consider my approach to job hunting. I went about it the way I have had success in Canada, instead of really getting to understand the system in Sweden, such as no dropping in without appointments.
3. You don't get anywhere in Sweden without knowing someone who knows someone, especially with apartments and jobs, and all kinds of opportunities. One thing I did that I didn't stick around long enough to reap the rewards was work with the government operated work placement offices. In Goteborg there is an office downtown that I went to a few times. They weren't very helpful right off the bat, but probably had I been more explicit with requiring more hands-on help, it couldn't yielded something positive. My one complaint with them was their tactic of sending you away telling you to go job search and report your findings back.
I would suggest asking everyone you know quite seriously to inquire about internships, temporary or part-time work, anything to make connections that will yield an interesting, more solid opportunity later on.
4. Swedes take ages to really become close to you, but once you are close they are life-long loyal friends. To this I can attest.
5. Swedish is a really hard language to learn. I'm taking courses again in Swedish back in Vancouver now, and ahhh!
Can your partner's parents or siblings/cousins/friends help with your work situation by taking you on temporarily or super part-time? I was able to work with my partner's mother at her school 'teaching' English to young teenagers (who teaches English when they can't speak Swedish??), which was an amazing opportunity, although the commute to the school ate up half of my pay.
This wasn't the most helpful reply, nor the shortest. I'll consider you when speaking to my Swedish friends in Goteborg.
Wish you all the best there, hope you stick it out longer than I did!
Monika