The fact that these offenses are defined as misdemeanors under state penal law is irrelevant: the definitions of crimes that can render you deportable or inadmissible are determined under immigration law, which is very different from the state penal law definitions. Immigration law defines a "conviction" as any admission of facts constituting the elements of the offenses, whether you are convicted by a criminal court or not, so the very first step in analyzing the possible impact of criminal history on your immigration case is to bring your court disposition records to an immigration lawyer, and have them look at the specifics of the state laws under which you were charged.
The real questions in your situation are generally,
1) whether either offense, or both taken together, make you inadmissible - and the marijuana possession charge probably does even if it is for simple possession of 30 grams or less, because any controlled substance violation within the statutory 3yr/5yr period is a bar, and it does not qualify for the petty offense exception since you have a prior conviction for something else. This is a temporary bar and just means you need to wait the 3yrs/5yrs until this offense is past before you apply.
2) whether either offense or both of them taken together could make you deportable (removable) and thus constitute a good reason not to apply for citizenship, as that would pose a risk of putting you into removal proceedings and losing your green card.
This latter possibility is more serious, because you cannot just wait it out: they have discretion to deny naturalization and find you deportable at any time if they find that you are a "drug abuser or addict" : to overcome this, you would want to first postpone applying for citizenship until both offenses are outside the required 3yr/5yr period for good moral character, and then get proof of rehabilitation including a physician's report or psych evaluation to establish that you do not have any substance abuse problem, that you have reformed your behavior and do not have either a chemical dependency or pattern of behaviour that could pose a threat to your own health & safety or that of others -
(the former being a ground for deportation and the latter being a ground of inadmissibility)
Reform/rehabilitation is a key element, and it is a totally discretionary call on the part of the natz examiner who interviews you. Natz examiners are very aggressive in looking for reasons to deny cases and/or find the applicant deportable, so a conservative strategy is a safe one.
In other situations, a concern that arises is the separate ground for removal/inadmissibility if you have two or more convictions for crimes involving moral turpitude, but simple possession isn't a CIMT, and a DUI isn't usually a CIMT unless you did it when your license was already suspended.
Bottom line: I would recommend that you apply for naturalization when you know you can get it, and meet with an immigration lawyer beforehand to get a careful analysis of the sections of state law under which you were charged. You will need to provide copies of the disposition records when you apply, and show the originals at the time of interview.
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