My well water smells and tastes bad. Sort of like petroleum. It also leaves a film in the toilet bowls that is white and grey. Originally I got enough 'sand like' material in the cold water feed for my washing machine that I had to wash the screen out in the water supply every 5 loads or so. The water also had iron. The landlord put a water softener in, but the smell/taste and film have never gone away. I want to get the water tested, but I don't know what to test for.

The house has a newer holding tank. It sat vacant after some plubming updates were made, and the holding tank wasn't actually hooked up when I first moved in. Someone came out to do that.
I thought at first the smell and taste was from new solder and it would go away in a bit, but it's been a long time.

I had an older bird fall ill, and I believe it's from the water. We no longer use the water for anything but showers. I'm so exhausted by hauling gallon jugs of water for a family of three and pets! I don't even water my plants anymore with it, so many of them died!

Take a sample of the well water to the county and have them test it. You don't have to tell them what to test it for as they will tell you what's in it. Then depending on what is in the water the landlord may have to shut down this well totally.

You didn't indicate where you are located. There could be a natural pocket of oil near the water well and it finally broke through and into the well water. Lots of factors at play here.

Where I live I can't drink the well water which is run through a water softener. There is too much salt in the water that causes me health problems. I have an artesian mineral spring 2 miles from my home that I go several times a month and fill up water jugs. I also give my little dog this mineral spring water and she's healthy.

It may be inconvenient for you right now but if the well water tests out unsafe to drink then what are you going to do if the landlord does not correct this problem and you have to move elsewhere?

And yes, birds are extremely intolerant to unpure water and will die from it.

Your landlord may not have the money needed for a new well to be drilled at a different location on the property. It can run into the thousands of dollars.