Would the 70/300 be enough?
That depends on how close you are. As a reference point, this photo was taken with a 70-300mm at 300mm from about 70m (200 feet), and is cropped to 1600x2000 from the 4000x6000 sensor size:
In general, zooming with your feet is a valuable technique. With wildlife, this often means using concealment to get close enough (the example photo above was taken from a hide), or picking a spot with a good view and waiting for the wildlife to come to you. Part of the technique is learning where in your area is a good place to get close enough to your subjects.
(Important side note: don't get too close to nesting birds. Showing attention in a nest can cause the parents to abandon it).
Also, do I need a lens with an internal auto focus motor?
If you want auto-focus, yes. If you don't mind learning to manually focus, that opens up your options for cheap long lenses; in particular, mirror lenses. A 500mm f/8 is easily in your budget, and if you look around you may be able to find a 500mm/5.6, a 650mm/8, a 1000mm/10.5 or a 1100mm/11. Note that the latter two are quite heavy and difficult to manipulate. But manual focus will mean a lot more missed shots, and in particular "bif" (birds in flight) shots are very difficult with manual focus.