Welcome.
With your budget you can afford just about anything you want for a G2 or Contender, including having multiple barrels scoped.
Personally I recommend finding a used Contender Easy Open frame in mint condition over the G2, and with either Pachy Grippers or custom wood made to fit your hand that will both help with recoil. And going with either 10" or barrels with a quality scope depending on which balances the best for you. A 10" is plenty to 100-125 yards though with most cartridges. So is a 2X, 4X, or 2-7X scope - Leupold and Burris are good choices.
If you want to stay with factory barrels, the 256WM, 25-35 Win, TC/U's (6, 6.5, 7mm), 357MAG or MAX, 30 or 357 Herrett will all get it done for deer and are easy on the recoil. Biggest problem with some of these will be finding them available for sale used. The 30-30 & 7X30 as well but with more perceived recoil maybe. For turkey maybe a 22MAG, 22 Hornet or a straight .410 Shotgun handgun barrel (not 45/410).
Barrel/caliber wise the skies the limit if you opt for custom barrels to hunt deer, especially if you reload. They can be had for about $300 new unbraked all the way up to near $700 new braked depending on who you have it made by. MGM is probably the cheapest, SSK the highest, Bullberry in between but has a way too long lead time. There are too many possibilities to list that would be plenty tame for you for deer without narrowing it down a bit to max range, and whether you prefer 24, 25, 26, 27, 28 or 30 caliber for example.
Some food for thought anyway...