Yep, two wrenches will do it for just aligning the existing. Two so you can break and tighten the nut without twisting the rod. Just make sure you fully understand which way you need to move it, like the article says each car varied (well, it's more where the tie rod is, front or back of the wheel). And move it only a little, it doesn't take very much to make a big difference. A quarter of a turn in or out might be all you need.

However, check the tie rods conditions themselves at the ball joint by the tire. If that's bad, that's certainly your problem, as that introduces lots of movement that shouldn't be there. Replacing it also isn't that hard, but to avoid a full alignment needed you have to try and match the new with the old in length as close s you can, and then are going to have to find where the old one was before the ball joint got bad. So a bit of trial and error and measuring there, more than if you just adjust the old. But it's doable. You're already ahead of a lot of people by knowing how to change your own oil.