Trading duplicate copies of your lux to the AI, or even buying their lux outright for 9gpt helps with your happiness (sell horses and iron as early and often as possible for 2gpt each, it will help you buy their extra luxes later).
Also keep an eye out on city state quests, particularly the ones that have luxuries and other resources you are lacking. Mercantile city states have their own extra luxury which doesn't exist on the map (like jewelry, porcelain), plus a normal luxury in their borders, PLUS they give you happiness as the bonus for friendship/alliance. So allying the right mercantile CS with luxuries you don't already have is a ton of happiness.
You can tailor your gameplan also. With basic 4-city Tradition play it's ideal to have 5 luxuries, and okay and totally normal to have just 4. But it's still possible to make 4 cities with only 3 luxes, you just have to grow slower and go for coliseums earlier, like before libraries even. If you have a good faith generation pantheon you can then help out by getting extra happiness via a religion. Or you could do a 2-3 city start and play taller, or a 2-3 city start and plan on taking some neighbor cities and their luxuries via war, or whatever. Stuff like that can make the game interesting.
It sounds like what you're looking for, though, is a different sort of map as opposed to a different sort of strategy. You might want to google "Hellblazers Pangaea map" in that case. AFAIK what it does is give each starting location a "regional" luxury, which has 5 copies nearby, but also provides more good expansion locations near you with unique luxuries.
It was designed for multiplayer games, going hand in hand with the "No Quitters" mod. The mod changes a ton of stuff like social policies and pantheons/religion, but with the map alone you should get what you're looking for in terms of reliably finding good expansion spots for yourself. Keep in mind the AI is going to get tons and tons of good expansion spots though as well, and they are way less penalized for making a million cities than you are.