Step one to solving this issue is figuring out the underlying reasons for it. Alignment is probably one of the worst mechanics in D&D just because it always leads to so much trouble. So lets start there and work our way up to your table.
Free Will 101
The first disconnect I see is in describing your character's alignment as "Do anything I want". The fact is, every character can do whatever they want already. That is literally the whole point of the game. All alignment can really tell you is how your character behaves in the broadest sense possible.
Law vs Chaos - Chaotic alignment does not mean you are a spaz that does any random thing that pops into your head. It means that your character believes personal freedom is more important than being constrained by a set of boundaries. Those boundaries can be personal beliefs (I will never steal), man-made laws (stealing is illegal), or divine mandate (my deity loathes thieves). Lawful characters are more likely to apply those rules to the situations they find themselves in, Chaotic characters are less likely. What actions a character takes is very broadly and generally affected, but it is not an absolute determinant.
Good vs Evil - The second axis for alignment is a lot harder to really nail down, since what constitutes Good or Evil in any situation is subjective (probably). But very broadly speaking being kind, generous, honorable, and peaceful can be considered Good. Being mean, greedy, dishonorable, or needlessly violent can be considered Evil. Again, it is all going to be super specific to the situation you find yourself in. Neutral characters do not feel strongly one way or another. This is the most important point to make as it applies to your character. Unfortunately it means you have to take an even closer look at every single situation to figure out how you would react.
You Are Probably CN (or close enough)
You, the actual person reading this and playing the game, are most likely to be True Neutral with maybe a bit of a lean towards one axis or the other. And that isn't a bad thing. I am sure that you think murder is wrong (Good v Evil) but also don't mind jaywalking or speeding on the highway on occasion (Law v Chaos). This means that you are close enough to your character as far as alignment goes that you can make an informed decision on how they feel in a given situation. Your character can have different motivations and driving forces, but from a pure alignment perspective you can go with your gut and be close enough.
Alright, now that that large digression on alignment is over with lets look at your actual actions in game and why you would or wouldn't do something. The scenario you described was the party walking into a shop and you debating the merits of assaulting and robbing the shopowner.
Now, as previously explained, your character is free to make any choice they want to in any situation. So on the surface this plan is feasible. You are stronger than the shopkeeper and could use those nifty magical items. But lets look at why you might not want to do so.
Societal conflict - Assault and robbery are pretty big deals in a city, especially if the merchant is well known. So going with your plan has a chance of bringing the local police after you. You mention blaming the robbery on Tiamat cultists as a way of deflecting this. But I would suggest that its just as likely this backfires and your party gets branded as cultists. Now you are doubly in trouble with the law.
And even if all of that pans out there is still a chance for ripple effects making things harder in your future. Maybe other merchants close up shop or hide away their expensive (high quality) gear for fear of being robbed. Maybe the cultists step up their timeline because of the increased pressure on them. Every action has a consequence and you have to balance whether they are worth it in the long and short term.
Intra-party conflict - It sounds like the other PCs in your party are a little more uptight about doing Good and obeying the Law than your character. They are very obviously going to have an issue with this plan. In their eyes you are breaking several laws and harming an innocent merchant just because you are greedy and have the power to do so. That sounds a lot more like Law Evil behavior than Chaotic Good (even though PCs aren't strictly aware of alignment in those terms).
This is going to be fairly obvious to your character, so the question then becomes is it worth the hassle to upset other people in the party just to get a few shiny artifacts? Whatever reason you have to be in the party probably outweighs the short-term gain and would be a big factor in how you act in a situation.
Intra-player conflict - Finally we come to the big issue, the fact that what you want to do in game is causing friction with the other players. Balancing why a random assortment of PCs are working together is always a bit of chore. It becomes an active pain point when things like alignment and actions cause people to butt heads.
Players don't want to break up the game, so someone has to justify why their character would put up with another character that they feel they would normally never associate with. Eventually someone has to reach a compromise or the group as a whole is going to suffer. That compromise can be anything from finding a middle path everyone agrees with, a character sneaking off to do stuff offscreen, or someone grudgingly accepting an act and holding a grudge about it (hopefully in-game only).
Free Will 102
To bring it all home, you have just as much free will as your character. In other words you can make any choice you want in this situation, as long as you are willing to accept the consequences. It sounds like you understand that some of the in-game behavior is causing problems at the table (which is kind of rare) so here are a few suggestions on ways to progress.
- Talk it out - This is the most important point. Tabletop games like D&D are collaborative experiences. So, talk to your group about what your character really wants and how you want to achieve that. "My guy is pretty greedy and is going to look for ways to make money. He also wants to stay in the party for X reason, so I will try to balance those two things."
- Pick your battles - This is an extension of the first point. Sometimes you are going to want to do something and you are going to be in the minority at the table. Figure out how important that action is to you and then either accept it isn't worth it, or explain why it is important and continue to push for it. Robbing the shopkeeper in your case is probably in the first category.
- Think about the long-term - A big point I make above is that your actions have consequences. You should not let that stop you from having fun in D&D, but you can use it to justify why your character doesn't do a thing which would cause problems with the other players. This way you can still get into character without it being an issue. "Normally I would seriously consider robbing this place blind out of principal. But I don't think Johnny Dogood would appreciate it, so I just quietly case the place instead".
Wow, this one kind of got away from me, size-wise. The super abbreviated version of all of the above is: Talk it out with the other players, and don't ever forget that the game is about everyone having fun. Doing whatever you want is selfish, and it is good of you to recognize that. It is just as true for you as a player as it is for your character. And just like your character you have the power and the incentive to choose less selfish actions in the short-term, if it means you get more benefits in long-term.