Take a look at your visa, you will see a enter before/valid until date, if that date is in the future you can leave Thailand, come back and get 1 more year of stay OR until the validity date of your health insurance.
OR you can go to your local Immigration Office and apply for a 1 year Extension of Stay, please ask them for their requirements.
It is almost impossible to give advice based on I have a Non O-A visa and I want to renew. You can not renew an issued and used visa, you can OR use the multiple entry, apply for a new visa OR apply for a new Extension of Stay.
You applied for a new visa using a school documents. If you not going to work at that school you should inform Immigration and cancel your application.
That also means you have to find a new and other way to be legal in Thailand.
Their must be a person at the place where you follow your internship that speaks Thai, let them call the Immigration Office and ask for the hand out with requirements.
The did manage to get all your documents together to apply for a Non Immigrant ED visa, it is not that hard.
Single Entry visa has nothing to do with the requirements of extending the period of stay you got on entry with this visa based on following an education or internship.
Strange that you're enrolled in a 6 months internship and the school can only provide a 3 month stay, what is the reason why you can not extend?
Alternative is go to a neighbor country and apply for a Single Entry Tourist Visa which gives you 60 days on entry and can be extended once on basis of tourism with another 30 days
The Visa is a personal item, which means each of you has to meet the requirements on their own.
You might look into a Non Immigrant O or O-A for one of you where the person must meet the financial requirements and for the married partner a Non Immigrant O as depending on the partner without financial requirements.
As long as you enter before the valid until/enter before date mentioned on the visa, you get maximum 1 year of stay OR until the valid until date of your health insurance, which ever comes first.