Mariah Carey did not need a nightcap the other night at One Little West 12th. According to a witness, Carey — whose album, “Memoirs of an Imperfect Angel,” is just out — showed up to the restaurant at “1:30 a.m. with her brother and a few bodyguards and was quite out of it. When she exited her black Maybach, she tripped twice and had to grab hold of one of her guards so she wouldn’t fall. She ordered a bottle of champagne but she could barely speak.” A source close to the singer tells Page Six, “She was probably wobbly because she was wearing 7-inch Louboutins. She had just completed 18 hours of press. That’s tough to do.”