Is there a difference between these two? Which one is more common?
I was wondering if I ( needed / need ) to take my own towel.
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Strictly speaking I think this is probably a duplicate of Past tense for indicating long distance (a somewhat misleading title, since the OP is actually asking about the significance of the verb tense in, for example, "What is/was your name?").
The answer is that in almost all contexts, the verb tense in OP's example sentence has nothing to do with Past/Present. Native speakers frequently use "I was wondering..." instead of "I wonder...", because the past tense metaphorically "distances" the speaker from the utterance - which has the net effect of being more deferential, hesitant, polite, formal.
Partly for the same reason, and partly just because it sounds more "consistent", we often describe "the thing wondered about" in the past tense as well. Hence...
1: "I was wondering what your name was"
usually means exactly the same as
2: "I wonder what your name is"
...but to most native speakers, #2 might often seem rather brusque, if not actually rude.
The present tense would be used in the following situation:
I'm going to the pool in my hotel. I call the front desk and say:
"I'm going to visit the pool. I was wondering if I need to take my own towel or if there are some there to use."
Now, for the past tense... well, it's the past tense:
My spouse and I get down to the pool and he says, "I heard you on the phone up in the room. What was that about". And I say:
"I called the front desk because I was wondering if I needed to take my own towel or if they had some down here."
Now, to add to this... From what Khan said,
I wonder if I need to take my own towel.
This is valid but it would be more of an internal dialogue. It can be said to someone else but is normally done rhetorically.:
I'm pulling together my stuff to go down to the pool. I think to myself (or wonder aloud to my spouse):
I wonder if I need to take my own towel... maybe I should call the front desk and ask
I am wondering if I need to take my own towel.
This is a slightly unusual statement that I would argue would be used rarely... specifically in a case like:
I'm sitting on the bed in my hotel room, thinking. My spouse comes into the room and asks, "What are you thinking about?". I would then say:
"I'm going down to the pool and I am wondering if I need to take my own towel. I think I'll call the front desk and ask."
Using "I was wondering if I needed to take my own towel" can be more polite. A listener doesn't need to feel compelled to accecpt it or handle it right now. I would use this sentence rather than the one using present tense.