I'm not familiar with Michigan law on the matter and unfortunately, there are too many different gun laws throughout the US to give good advise without several hours of reading your state's laws.
But in my state, Kansas, it is NOT "illegal to posses a firearm while intoxicated." It is illegal to carry (concealed or open) a firearm off your property while intoxicated.
Inside your own home (in pretty much any castle-doctrine state) you can shoot any intruder, whether you are drunk or not.
If 'a reasonable person' would be in genuine fear of their life in your situation, then you can not be arrested in your own home for the self defense either (In Kansas anyway).
Law and court precedent also provides (in castle doctrine states) that anyone having gained unlawful access to your domicile, may reasonably be assumed to be a deadly threat to you or your family. They don't have to be aggressive to you. They don't have to have a weapon. They do not have to have used force to break in.
If you invite them in that's another story. But if you were not aware of their entry, or their entry was against your command, then they are legally assumed to be a threat to your life.
There have been cases where an alzheimer's patient, curious child, or drunk teen has entered the wrong house via an unlocked door, been shot, and the homeowner is not criminally liable, at all. (Though a civil lawsuit could always follow)
Owning a firearm DOES NOT MEAN you have to swear off all drinking, forever. In fact in many states, you CAN drink while armed, in public. But you may not "be intoxicated", and whether you are or aren't intoxicated is going to be a subjective observation of an officer, who is certainly not intoxicated themselves, and probably very familiar with signs of intoxicated persons, so I wouldn't recommend drinking any more than a tiny fleeting sip if you're armed.
Very few states set a BAC limit for firearms possession in public. It's either NOTHING IN YOUR SYSTEM AT ALL or NOT INTOXICATED. And frankly public intoxication is almost always a crime anyway whether you're intoxicated or not.