"My wife and I am" = one family, a singular number. Therefore satisfies the use of "am" for both number and proximity issues.
People don't normally perceive "my wife and I" as singular.
There are many situations where you can use a singular verb with a plural subject to make the listener think of the subject as singular. Here's an answer that gives a few illustrations of that. That doesn't work with "my wife and I", since even though those might be one whole family, the word and reinforces the perception that they are two people. In British English, even the singular noun "family" often takes a plural verb.
Regarding proximity agreement, where the verb only agrees with the part of the subject closest to the verb and disagrees with another part of the subject, that works with or, not with and.* The main reason why proximity agreement is needed is because part of the subject is singular and part is plural, so there is no way to make the verb agree with both parts:
Bob McClaskey is singing tonight.
The Three Irishmen are singing tonight.
Bob McClaskey or The Three Irishmen are singing tonight.
The Three Irishmen or Bob McClaskey is singing tonight.
So, if you're both singing:
I and my wife are singing tonight.
If only one of you will sing, and you haven't decided who yet:
I or my wife is singing tonight.
The verb form am suggests so strongly that the subject should exclusively be "I" that people tend to reword the sentence rather than use proximity agreement to cover a compound subject including someone other than "I". For example, if you reversed the order of "I" and "my wife" in the previous example, you would probably say:
My wife or I will be singing tonight.
to avoid the disorientation that results from "My wife or I am singing tonight." The verb form am clashes too strongly with "my wife" for proximity agreement to make up for it.
* That's not a rule, of course. Here's a question about a sentence where proximity agreement does work with "and".