I am afraid that your request is so general as to handicap any efforts to help. For example, not only will a Saints game pack just about everywhere in sight but, also, the French Quarter is walking distance so there could be a LOT of people out eating beforehand (although once a game starts some places are rather pleasant). October is too early for crawfish. We tend to see pond crawfish sneaking in around late November or early December .
You may as well ask for the One Authentic Cassoulet Recipe as ask for The Best Po boy. The categories subdivide rapidly: Oyster/shrimp/roast beef/meatball/. Then there is the fill-in-the-blank variety that we've had lots of in recent years. Just put something on the bread and run: soft shell crab;frogs legs; "duck sliders" it goes on and onto into a mass mania.
As HH points out, there are so very many options, that it is difficult to offer suggestions.
A Saints game sort of clouds the issue, due to the crowds around the stadium, and well into the FQ (French Quarter).
Let's take that out of the mix early. Where are you staying? Do you have an auto, or will the somewhat limited public transportation work for you? Do you mind walking?
For other days/nights, beyond the game, here are my suggestions, but there can be many other options:
Galatoire's for a wonderful NOLA dinner. For gentlemen, a jacket is required. If you have not done Galatoire's, this is a must.
Po-boys: wow, this is a broad category, but here are my two favorites, and might require a streetcar, and then a little walk: Roast Beef - Parkway Bakery and Tavern. Shrimp, or Oyster, then Felix's in the FQ. However, those are not even close to the only options.
Game day? Well, it would be whatever you can book, and are somewhat close to.
Many folk ask me what to see in NOLA, during Mardi Gras. My only answer is, "see Mardi Gras, and when you want to actually see NOLA, visit at some other time." Saints games, Sugarbowl, Superbowl, etc., can be somewhat similar. They are the events, and one must sort of work around them. During such events, the City is "not your oyster." There will be crowds, and many will have booked tables well in advance. Luckily, many of the "fine-dining" restaurants will not be in THEIR mix, so you can usually pick and choose, just a bit. More relaxed, less-formal dining, will likely be slammed, so pot-luck will be the course of the day. The farther that you get from the stadium, the easier it will be to secure reservations.
I do not want to rain on your parade - actually just the opposite, but want to point out that much of the area around the stadium can get filled, and maybe even months in advance. Much of my wife's family, will head to near the stadium, and dine early, or late, before a game (been that way, since the first days of the Saints), and much of NOLA is that way. Depends on their time, and their schedules.
With an auto, one can perhaps dine at Ralph's On The Park, early, then head to the game. Same for the Windsor Court's New Orleans Grill - early, or late, then fit the game in.