УЖЕ и УЖ sometimes do have the same meaning. When they do use as an adverbs. E.g.: 1) Here I am (I'm ready) - Я УЖЕ (УЖ) готов. 2) It is a long time since - УЖЕ (УЖ) давно. Here is an example below which shows the difference between УЖЕ и УЖ. УЖ sometimes is used as a particle in BE SURE form. E.g.: 1) He is sure to do it all - УЖ он всё это сделает. 2) He is sure to find out everything - УЖ он всё узнает.

I'd like to add to Aidar's comment that in sentences similar to his first 2 examples, 'уже' is preferable in the modern Russian language.

Let me add some complementary examples for “УЖ” and “УЖЕ”: 1) Если он так уж сильно хотел прочесть это, он должен был бы здесь. If he wanted to read it that very much, he should be here already. 2) Лучше уж я пойду погулять! I had rather to have a walk! 3) Не так уж давно это было. Not so long ago it was. 4) Уже пора! It’s time to go!

Definitely not. "Уже" can be used in different meanings, and its using differs enough from "уж". There are many cases when you cant use "уж" instead of "уже" and vice versa. Meanings from dictionary: УЖЕ - already; by this time, by now; now; no longer УЖ - certainly, definitely; to be sure, really You can find examples in lingvo.yandex.ru just type уж and уже in search form Though, often УЖ can be just shorted version of УЖЕ, but it happens mostly in oral comminication not in writing.

There is another interesting expression with уж https://www.italki.com/question/377059