If you do not live in the US, you are not liable to state tax but if you have a business in the US, it is still “resident” there, so it is still liable to pay state tax. You moved, the business did not.
I understand that first you pay the taxes of where you live (i.e. the foreign country taxes), then you use the treaty to deduct whatever is possible when you file the USA form. You do not need a tax adviser for that - simply ask the IRS, they provide detailed instructions on what you have to do (or are supposed to do).
If you live in a country where you are liable to substantially lower taxes, you can renounce your citizenship, but you should think about it carefully before you cross that bridge.
On tax avoidance, there issue is that there are myriads of situations and each one has a different approach - in general it depends on individual cases and are only worth the trouble if there are reasonable sums of income involved or if you want to avoid being liable to tax on specific income (inheritance, for instance). In your case, you may find advantageous (or not) to shift your US based business to another place that is more favorable. Check for ‘tax havens’ in the internet. The problem is that you are required to declare all foreign based firms that you have when you file US taxes, which is how they catch you by your feet. The point is evaluating if it is worth the trouble and risk.
It may be, it may be not. Not forgetting that you will also have to deal with the taxes of the country you reside.