Avoid Badalona, it is a very low district. You don't move to Barcelona to live there. The highway goes through it, so what I have seen is from the car.
I second Gava and Castelldefels. There are very close to BCN and really nice,
We lived in the centre and 6 years later, I still miss Barcelona so much and I am in Sydney now. Very happy place, gorgeous city.
The only BIG negative are the pickpockets. When we were living there, we were members at the club de natacio in the barceloneta. www.cnab.cat Its was very cheap and it meant we would have a locker to leave bag, and the pools, and access tot he beach in all safety.
Once you know about the pickpockets, and you know you have to leave your bags on lap in cafe and not on floor or back of chair, crossbody bags when walking around , not leaving a bag on the beach when you go for a swim, .....you adapt your behaviour and it is fine.
We first were living at the top of Rambla de Catalunya (different from Las Ramblas, other side) and I loved how vibrant, elegant it was. We would walk to Gracia for ice cream, everything was there. We moved to Vil-la Amelia to be closer to the kids' school (French Lycée) but I so much more preferred the city centre.
You do know about Catalan , right? Many will refuse to speak Spanish to you, even if you are a polite foreigner making an effort. My paediatrician was one of those, and our encounters were ridiculous, I would speak in Spanish, she would reply in Catalan and so on. A couple of librarians, and other people here and there in the public administration when you needed some random paper.
I did both Spanish and Catalan courses at this school www.eoibd.cat/ca/ the city official school. You don't need Spanish obviously, but doing a Catalan course will help you connect with more people.
If you are feeling "flat", Barcelona will recharge you in a second.
It is beautiful, happy, chiringuitos on the beach, happy faces around you. When we were living there and would visit family in France, my youngest would ask me " why is everybody so angry" when walking in the city centre,.
Nobody would say " bonjour mon trésor" when you enter a bakery, but the " Hola cariño" is so nice to hear.
Work situation is however dreadful but you could maybe give English classes,