We will be staying on an e-tourist visa in India starting next week and would like to extend our stay.
Is it possible to apply for a full tourist visa from within India? If so, how?
Normal tourist visas can only be gotten by handing an application into an "Indian Visa Application Center (IVAC) or directly to [an] Indian Mission." These facilities are all found abroad, you don't find them within India. Moreover, for your visa to be valid it also needs an entry stamp from an Indian border official. The only way to acquire such a stamp is to enter India from the outside.
So, unfortunately, it's not possible to get a new tourist visa while still in India.
You cannot extend an Indian E-Visa and although confirmation of the visa is generated when the online application is made the visa itself is issued and starts when the applicant enters India via one of the airports listed so it is not possible to apply for an E-Visa from within India.
Source. Emphasis mine.
You would have needed to apply for normal tourist visas before leaving your home country. These applications have to be printed out and physically submitted to an actual Indian Mission or IVAC. These visas can have a duration of up to six months. Unfortunately, if you're traveling in less than a week, you might not have time to do this. I've found that the processing time for these can be anything from two to ten office days. (Sometimes more, if they want to question you, etc.)
However, you could contact the Indian foreign mission serving your area, explain the situation and ask if it's possible to get six or three month tourist visas issued on a short notice.
Another option: When your e-visa is up, you could travel up to Kathmandu (or even down to Colombo or basically any country you'd like) and get new visas from there, then return to India. Please note, though, that this will not work for citizens of Afghanistan, China, Iran, Pakistan, Iraq or Sudan, or stateless people. Such travelers are required to observe a two month gap between visits to India. However, all other nationals can return immediately once a new visa is granted. I did this in 2012. My six month tourist visa was about to expire, so I flew down to Colombo from Chennai. I immediately applied for a new six month tourist visa, and I had it after about seven office days, if I recall correctly. I then returned to India, after around ten days in Sri Lanka.
Yes, such a trip seems like a waste, and it's frustrating that this is the way it is, but unfortunately, those are the rules, and there's really nothing else to do. And if you do it, you can enjoy the experience of the neigbouring country while waiting for your visas.