I appears that he should be able to adjust status under section 245i of the INA, since he is the beneficiary of a petition filed before April 30, 2001. All this assuming that you are a United States citizen and that he is otherwise eligible to become a permanent resident (For example, if he has committed a crime that might or might not affect his eligibility for adjustment of status).
A tourist visa is a nonimmigrant visa that allows you to come to the US to visit and then return to your country. If you came on a tourist visa knowing that you were going to marry an American citizen and then apply for a green card that would be a violation of US immigration laws. However, if you came to the US to visit and later on your plans changed, then you might be able to argue that you did not violate any immigration laws since your original intent was to visit the US temporarily. It...
It's going to depend on your legal status in the US and actual charge you get. Generally, if an individual is convicted of drug trafficking offense, that person may be deported from the US. More facts (date of the plea agreement if there's one, etc) would be needed to determine the outcome of this particular case.
Generally, if you overstayed your visa and you're married, to get a green card in the US without having to leave the country, your husband would have to be a United States citizen or you would have to have a United States citizen son/daughter over 21 y/o who could petition for you.
When someone spends more than a year in the US with no legal status and then departs the country, that departure will trigger a 10 year bar. However, there's a waiver available, which, if approved, would allow him to come back.
If you're an American citizen, you can file a petition for him. Generally, when someone comes to the US with a visa, that person can adjust here in the US if petitioned by a USC spouse. If he meets other legal requirements, he might be able to become an LPR in the US.
Are you a US citizen? If so, based on the facts you described, he won't have to leave the US even though his visa expired. You're supposed to make enough money to support him based on the poverty guidelines. If you don't, you'll have to get a joint sponsor. As I always suggest, talk to an immigration lawyer.
If you came to the US legally and you're married to a US citizen, you should be able to apply to become a lawful permanent resident, assuming that you meet all legal requirements. If you left the country and came back illegally, some bars might apply. The same if you have a criminal history.
You should talk to an immigration attorney.