Ok Bob, I think I know what your problem is. It is one of two things depending on your description. #1. A high irritating sound when you hit your snare. If it is a pinging sort of ring, that can be solved by placing a zero ring on the snare drum head. Nothing fancy. All drums are a combination of sound. High pitches are to the outside toward the rim, low pitches are toward the center. Eliminate or reduce either and the sound of the drum shifts. Try that zero ring and you should get a drier punchier sound. Be warned, you will lose volume as it is the high pitch that cuts through the band. If you use mics it won't be an issue. #2. The snare rattles after you hit it and you can't get a pure tom sound. Take a deep breath...your snares are too tight. This is a real common problem that I see almost every day. Nobody teaches us how to take care of our snares or how to effectively use the snare strainer. So here you go: First look at your snares. Are they still any good? If they are bent at all in any way shape or form replace them. In fact, if you have been overtightening them they are probably stretched out and thus lack the ability to snap. (Think of an elastic band on a pair of underwear. After the elastic loses its spring you can still wear the garment, but it just isn't the same!) Replace them. Snares aren't that expensive. Step 1. Remove the current snares. Step 2. Unscrew the snare strainer until you find the mid-point (or slightly lower) of the tension screw (what you have been using to tighten the snares). This is where you want to start to set the snare tension. Step 3. This is the tricky part. Depending on if you use tape or string the technique will change. I prefer string as it give me a more even set and less slippage which is what I think you have been experiencing. When you put stress or tension on the snares the snare butt plate is unable to hang on to the slick tape used to hold the snares. Snare cord works much better for me. Loop the string through the holes in the snares' plate. Run the cord through the butt plate. Tie a square knot (Boy Scout Handbook). Tighten screws into place. Step 4. Repeat Step 3 on the strainer. Step 5. Now the tedious part. You want to center the strainer against the bottom head when the snares are turned on. To do this take a drumstick and place the tip end through the loop sticking out above the buttplate. Turn the snares on. Loosen the buttplate screws and use the stick to pull the snares tight (like using a lever, you want leverage). Tighten the screws into place. If it is off-center, make adjustments in the cord on the strainer side as it should not be used with stick pressure and will be a more permanent setting. Repeat until you get it right. This will take some time regardless of your level of experience! Step 6. Turn the snares on. They will probably rattle. Without a second pair of hands it is nearly impossible to exert enough pressure on the stick, hold the drum, untighten and tighten screws, etc. to make it perfect. Step 7. Turn the Strainer Tension Screw until the snares are off even when the mechanism is turned on (a 'pure tom sound'

. Step 8. Using your one drumstick play steady quarter notes as you tighten the Tension Screw. You will start to hear the snares razzing. Continue until the razzing becomes a snare sound. Play at three volumes- pianissimo (REALLY soft), mezzo forte (normal), fortissimo (rock and roll loud). Continue tightening until the razzing stops at all three volumes. DO NOT GO PAST THIS POINT!!! By over-tightening the snares you actually choke the drum. The snares are too tight to resonate and the drum sounds like #!^*. Step 9: Tune the snare drum. Bottom and top heads (tuning is another issue for another time).
That's it. If that doesn't fix it, chunk it in the river and go buy a decent drum. A word of warning: Many people seem to think that the harder they turn the Snare Strainer Tension Screw the tighter it will continue to get. Wrong! Once the screw is out of thread, that is it. Continuing to turn just means you are breaking the mechanism. It is a machine. It won't heal or give an extra effort. All it can do is break. That is why Step 2 is so important. You should learn the capabilities of your equipment.
Thus endeth the lesson.
md (Matt Porter-for those who prefer names)