I live in Arizona, and I can never tell if I am watering my plants too little or too much. Plants are very touchy here - and you have to be able to read the signs correctly. I can't.
What signs are there for each?
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There are various ceramic tools you can use, but I've never tried them.
There are signs, tomatoes for instance, will have more fuzz on the sides to collect any water available
But most gardening books just say stick your thumb into the dirt, if it feels dry then it is time to water, don't wait till the plant turns brown.
That being said, make sure you don't water too much and have some system of making sure your water gets down to the roots (a little berm perhaps). That'll prevent your plant from keeping its roots up near the top of the soil where it dries out quicker.
I have a moisture meter with a 6" or so long probe. It's great for checking soil moisture below the surface, and it's not very expensive. I'm afraid to stick my finger in deeply and disturb the roots for some plants. Also, I've seen the top inch of soil be dusty dry, but there's a surprising amount of moisture 3-4" down.
Once you get in the habit of using it, you'll get a better feel for how much moisture is really in the soil.
This is much easier and more reliable than looking for signs of stress in your plants. And if your plants are showing signs of stress, they're already damaged -- better to avoid that in the first place.
Water less frequently, but with a higher quantity. Make sure the water gets down deep to the roots. (Make exceptions for thirsty plants with naturally shallow root systems, but I wouldn't be planting these in the desert!)
Increase your soil organic matter content. (Mulching can do this too.) Organic matter in the soil will hold more moisture for longer.
If you say your plant is very sensitive to over-watering, I assume it is a kind of plant that doesn't like a lot of water. The key is to know how much water your plant needs. Try googling for more info regarding your plant. The season also matters; in the summer you should water your plants more often, while in the winter you should reduce your watering.
My advice is to stick your fingers into the soil. If it is completely dry, you may want to water it. However, if the weather is not hot and the plant hates water, you can wait and see for a couple of days. Otherwise, you can water it when the soil is dry.
When you water the plant, you have to pour the water gently and keep watering until the water comes out of the pot. This makes sure the water gets to the root and not just the top layer of soil.