Simply widening the aperture, alone, may not be the solution if AUTO ISO lowers the ISO in a way that maintains the same overall dark exposure. You need to pay attention to how the meter is reporting the exposure, in the view finder telling you how things are going. These two posts have a lot of discussion, the second post has an editing example. Both posts show looking at the highlight warnings and assuring the meter is reporting an exposure 'to the right'.

ETTR in Practice

ETTR in Practice II

Try it. It's not brain surgery. When I got my first DSLR I worried about cleaning my own sensor. Eventually I got a new DSLR so I tried cleaning the old one. It was really simple, no problems were encountered, and so I cleaned the new camera also. Been cleaning my own since then.

Paul gave you a link to his cleaning post. Worth reading the thread. I have my own page on sensor cleaning. There is not one way to do it: different people have different methods. That alone should let you know that it's not a difficult job.

You need some things to do it. The Rocket Blower is frequently recommended. That is a brand name, but the basic idea is that you blow off any dust on the sensor. You just need a blower that you can keep clean. The Rocket Blower (or equivalent) is probably available at any photography store.

The Rocket Blower is sometimes all you need. But if you need more, you may wind up doing a wet clean. That requires a sensor swab. There are different sizes to fit different sensors. A photography store will be able to tell you what size you need for your camera.

Not bothering the animal is why we shoot with long lenses. You will also find that the distance that actually does bother an animal is a moving distance in some ways, unless defined by a national or other park you are shooting in. Animals that have been raised near or around people will put up with a shorter distance than an animal not raised near or around people. Over time you learn to realize when an animal may feel "bothered" and at that point you simply move away to a distance where the animal seems to feel no longer bothered. Just the other day I had two foxes under my deck at 10 in the morning..they sort of ambled off when I shooed them out...they obviously, unknown to me, are familiar with my house and me! I photograph a lot of wildlife, including with Steve Perry and some others, and it is a journey to get the shot you want but always well worth it. In three weeks I'm off to Yellowstone for the sixth time and hoping to get that elusive wolf shot, my fingers are crossed. In December I was at Bosque del Apache and in January at two wildlife refuges. Each setting is different with different animals but each is similar in how I shoot...f/5.6 with a long lens for the most part. If the background is clear, i.e., sky I can shoot at a larger aperture but often I just leave it at f/5.6 because the distance between my lens and the animals is so far away that it does not matter if I change the aperture one stop down. Good luck with your new avocation with wildlife...it can be addicting.

Simply widening the aperture, alone, may not be the solution if AUTO ISO lowers the ISO in a way that maintains the same overall dark exposure. You need to pay attention to how the meter is reporting the exposure, in the view finder telling you how things are going. These two posts have a lot of discussion, the second post has an editing example. Both posts show looking at the highlight warnings and assuring the meter is reporting an exposure 'to the right'.

ETTR in Practice

ETTR in Practice II