Imo I think you'd be much much better off trying to find a used Savage 10 or 11. They are pretty easy to pick up in that price range and quite a bit better than any of your options.
That said if you can't find one, or absolutely can't swing a new one I'd probably choose the Ruger American first and the Savage Axis second.
I had a Marlin XL-7 and I hated that rifle. I couldn't wait to get rid of it. I read many reviews about it's accuracy, and it was accurate. I rarely got a group over 1" at 100 yards, but the whole gun was super cheap feeling. The trigger was lousy and wouldn't adjust below 3lbs, the adjustment screw and trigger felt super cheap, they commonly break extractors, the gun was super hard on brass, the scope base screws for the factory base are very weak but it's hard to find any other screws that will work with that base, etc. I just didn't really like a single thing about it other than the accuracy. If it's all you could buy it would be one thing, but why buy it when there are soo many other great choices in that price range.
The Axis isn't a bad rifle, but the stock is super cheap on it. The Axis II has the accu trigger which is great imo, but the original Axis has a heavy trigger. It can be adjusted with some modifications, but imo it's just a huge step down from the accu trigger, so I'd probably get one of the Axis II's if I was going this route. The magazine setup is weak on it and cheap feeling, and the stock is pretty lousy feeling. Other than that it's a solid action and they are generally very accurate. I think these are good guns, I just think there are also better options for not all that much more.
The American is a great gun. It has a decent trigger, they are generally very accurate, and they are pretty solid guns all the way around. They aren't up to Remington 700 or Savage 10/11 quality imo, but they also aren't the price when new. In my opinion this is the budget rifle to get, but I just think you can get a lot more for your money if you spend an extra $75-100.