She doesn't want to marry Becky because their relationship was, and is, entirely predicated on Amber needing a surrogate parent to accommodate her and drive her to Food Places. Their relationship is totally mercenary: Amber is a vicious narcissist who plies Becky with money to drive her around so she can glut her tasteless tongue with cheap trash, and Becky is a lazy halfwit who knowingly enables Amber's piggery in exchange for men's graphic tees and crayons. Amber says she wants to move, but what she is actually saying is that she wants a better food slave in a better location: if Amber caught wind of another easily manipulated dimwit lesbo with a better resumé than Thumby, the next day Becky would come home to nothing but Amber's awful smell and several abandoned pets.
But, with Amber gaining weight faster than a bear in autumn, it may prove very difficult to dislodge herself from her current sconce. She is not the sprightly Amberlynn of her clapping.gif days; when she hits 600 next January she won't be able to stand up for more than a minute or two, and she couldn't move house in that condition. Smugly gorging herself with Becky is going to have a price, paid in living out the rest of her days with a misshappen butler who looks and sounds like Humpty Dumpty had a stroke.