Yes, you can move into a house on your own. The instructions are
here, complete with pictures. The only picture missing is the one for finding your sim to move her into a new house. If you're at the map, look in the upper righthand corner. There's an icon showing several people. Click on that to pull up played and unplayed households. You'll find your sim in the unplayed section.
Your sim will keep her relationships, skills, job, and whatever is in her personal inventory.
ETA: I forgot to mention that your sim will have only $20,000 to buy a new house with. Also, welcome to the Forum!