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Oaxaca, Mexico is a nice choice. It’s a beautiful location, lots to do, tons of culture and fabulous crafts, lively place both day and night. Here is their web site http://www.oaxaca.gob.mx/. If you don’t speak Spanish, you will need to translate the site.
Merida is another possibility. Yes, there are tourists, but if you walk a few blocks in any direction, there are fewer luxury spots and more basic hotels and eateries.
This is true just about anywhere you go in Mexico. You can fly into whichever airport you want and take a bus to just about anywhere. A few blocks and you are out of the tourist zone - even in Cancun and Mexico City.
Sorry, you hadn't mentioned that you wanted to go deep sea fishing. I was looking for family vacation destinations. If your focus is fishing you will want to stay near the water. Acapulco, Puerta Vallarta, Mazatlan, and Manzanillo are all on the Pacific. While these are typical tourist destinations, you can get off the beaten path and enjoy less expensive and more authentically Mexican hotels and cuisine.