You are right: 'мне грустно' is the most common case. Look at the word in different parts of speech (verb,adjective,adverb). Just compare the variants:
What are you doing? Request for action. Mostly used when speaking about someone else (using that for yourself is not very appropriate, it usually makes you looking lazy/weak - so speak about yourself in such way only with close friends):
Я грущу = I am feeling sad [now]
По-моему, он о чём-то грустит = I think he is sad about something
What person are you? What person are you now? Request for description. Obviously, you should not say about yourself in a such negative way:
Я грустен/Я грустный = I am sad [I am a sad person]
But in Russian it is assumed that sadness will pass somewhen. It is something that distracts you from your friends and your work only now. Sadness in this meaning is some kind of external phenomenon:
Мне грустно = I feel sadness
You will completely understand it if you look at sadness like at other externals (like 'it is rainy', 'it is snowing'):
Мне холодно = It is cold for me (I feel cold)
Мне противно = It is distasteful for me (I feel distastefulness)
Мне хорошо = It is good for me (I feel good)
Мне тяжело = It is hard for me
Regarding other expressions of sadness, I can suggest other common 'sad' expressions (same can be said about drugs):
Меня не отпускает грусть = The sadness doesn't release me
Меня внезапно накрыла печаль = The sadness has suddenly covered me