As for which would be worse, that's very hard to answer without a few examples to consider, so I thought of these:
I bought car on Friday. (should be: I bought a car on Friday.)
I went to store and bought chips. (should be: I went to the store and bought chips.)
Then, I tried to come up with examples where I put an article where it doesn't belong. I found this was much harder to do! Often, including the article doesn't make the sentence wrong, it simply changes the meaning. Consider:
(A) I went to the store and bought ham.
(B) I went to the store and bought a ham.
(C) I went to the store and bought the ham.
All three of those are correct!
Sentence A means that you bought some ham at the store (we don't know how much ham, or what kind of ham – it might be sliced ham, it might be a whole ham, it might be 25 hams).
Sentence B means that you bought one ham at the store (not a package of ham, and not two hams, but one ham).
Sentence C is usually used when the other person is expected to know which ham you are talking about. This might be because you're answering a question:
Where did this ham come from?
That's from me; I went to the store and bought the ham.
Or it might be because there was some context set in the past; for example:
What do you want to serve at party this weekend?
How about ham and turkey sandwiches?
That's a good idea, but we'll have to buy more ham. We're almost out.
I went to the store and bought the ham.
(In that last sentence, the inclusion of the word "the" indicates I bought the ham we needed, or, I bought the ham we talked about.)
Of course, here's an example when the article is just wrong:
I went to the store and bought a rice. (should be: and bought rice, or, and bought a bag of rice)
I don't think either gaffe is really any worse than the other. You'll just have to master it through experience. Hopefully, the people you converse with will give you helpful feedback – not with a patience and understanding, but with an attitude of patience and understanding (or simply with patience and understanding).
(We can talk about "my patience," or even say, "he has the patience of a saint;" still, no article was required in that last sentence.)