Is this stray cat sick or just starving?

My boyfriend and I live in a farmhouse with the nearest neighbors being about half a mile away. A few days ago, this stray cat appeared on our doorstep looking very very thin and had a hoarse meow. We've been feeding it, and lately it's been showing up with a very dirty face, like it's been drooling very profusely and then wiped it with dirt. It's cheeks appear to be slightly swollen and it's left eye has tear streaks. Like I said, we've been providing food and water for it, because it is bone thin. Before, it wouldn't let us near it, but lately it's been letting us get very close to it without showing signs of running. Does anyone think that it is exhibiting signs of disease, or is it just starving? I would like to take it to the vet and get it checked out and possibly vaccinated because we have other animals that live outside and we don't want them to get sick if it has anything. But my boyfriend thinks that it is suffering, and he said that if it doesn't seem better in a week, he's going to shoot it and end it's misery. I don't know what I should do for it other than just feed it! Help please! I hate seeing this poor animal every day and not knowing what to do for it :(