If you want to try correcting a tin ear at home, it certainly can't hurt, and some headway may be made, particularly if a loved one will play pitches, listen and give honest feedback. And if the breathing works consistently well.
About breathing: A common method involves taking a big breath and filling up your entire upper body, from below your waist to your shoulders, expanding not only your entire abdomen but also your whole back. This should be done with a straight back, either sitting or standing.
When you exhale, or sing, keep the back straight. No one area of the upper body should be stressed, but you should feel both the back and abdominal musculature working as a unit to push the air out of your body. Be aware of the individual parts working, but again, don't emphasize any one of them.
To most easily feel the muscles working you probably will have to push the air out vigorously, which will result in a louder sound than perhaps you are used to making. But don't let this deter you! As you gradually become accustomed to using your body this way, so, too, will you become accustomed to making a full sound.
If you are breathing well, the neck and throat will be relaxed -- even with a loud sound. So try to use that as an indication that all is going well.
Tension or tightness indicates either:
(1) Fatigue of the breathing apparatus, in which case you should stop singing.
(2) You are trying to go too high for what your voice can do comfortably with a full sound.
(3) You are too loud in a low range or too soft in a high range.
(4) The breathing apparatus isn't working properly and you should return to square one and start all over again.
Regarding pitch, if you merely have a slightly "rusty" ear that just needs some quick tuning, you may get on pitch by adjusting the amount of muscular pressure you are using to make the sound. If someone tells you that you're under the pitch you're aiming for, or flat, add a little more pressure, or volume, and the pitch should rise. Reverse the procedure if someone tells you you're above the pitch, or sharp.
When you are satisfied that you have matched a pitch and your helper confirms this, then begin to venture up a step and back down, then up two steps and back down, etc., going only as far as you can with correct pitch. Then vary the starting note, sometimes starting higher, sometimes lower, always expanding the range of your exercises according to the progress of your sense of pitch.
When your ear is fairly well along, a tape recorder can replace your second pair of ears.
Gradually you will be able to zero in on a pitch unhesitatingly with good body-and-ear coordination, and at some magical moment you will notice that your pitch is good without your having to think about it. Yes, you can get there. But if you start despairing in your at-home practice, you may have to call on a pair of trained ears to help you out.