"Well, I don't remember."
Excerpt from 'Japanese Words Their Uses II: The Concise Guide to Japanese Vocabulary Grammar: Learn the Japanese Language Quickly and Effectively: 2':
To express the idea of 'retaining something that has been committed to memory,' one has to say oboete-iru rather than oboeru. American students of Japanese often make the error of identifying oboeru with 'remember' and make sentences such as (3), but (3) is a misrepresentation of (4) and cannot mean 'I don't remember his name.' / (3) Ano hito no namae wa oboemasen./ / (4) Ano hito no namae wa oboete-imasen./ I don't remember his name. Sentence (3) can be correct only in the sense of 'I won't (i.e., I refuse to) commit his name to memory.'
Please remember, to express current states in Japanese, we must use the progressive form ～ている. Progressive form represents both the continuous action as well as states (where in English normally we use present or perfect tense to represent).
(私は)覚えます I am going to remember. / (私は)覚えています I remember.
(私は)覚えません I am not going to remember. / (私は)覚えていません I do not remember.
映画が始まります The film is going to start. / 映画が始まっています The film has started.
映画は始まりません The film is not going to start. / 映画は始まっていません The film has not started.
車が止まります The car is going to stop. / 車が止まっています The car has stopped.
車は止まりません The car is not going to stop. / 車は止まっていません The car has not stopped.
If you really want to say "The car is stopping," 車が止まりかけています or 車は止まる途中です.
I write helpful information in a notebook. Analog, I know, but it's easily accessible, makes me practice writing kanji/kana, and the extra time it takes forces me to pay attention to the information longer. Plus, it gives me a reason to use the notebook someone got for me as a gift.
覚えません would not be correct. 覚えていません is correct. 覚えていません means that one 'is not remembering', while 覚えません means 'to not remember'. ～ていません is the present progressive, while ～ません is more the simple present(?) / negative infinitive(?). Sorry, grammar is not my strong suit.
Actions which are current and ongoing are expressed using the ～ていません form. ならっていません (I'm not learning), べんきょうしていません (I'm not studying). Basically, it translates to 'I'm not remembering', but this is how 'I don't remember' is expressed in Japanese. If you were to say おぼえません it would mean that you don't remember things in general.
It's kind of hard to explain. I believe じゃあ is used at the beginning of an expression which suggests performing or not performing an action ('Well, let's go', 'We shouldn't go', etc.). さあ can also be used in this form, but has another use such as the one being used here.
'I don't remember' is not an action, it is a state (albeit a negative one). Therefore さあ would be appropriate.
On the other hand, if you wanted to say, 'Well, let's forget about it.' you could say じゃあ、忘れましょう。because it is leading to a suggested action.
Additionally, さあ can carry the nuance of a feeling of surrender or acquiescence to things which are beyond one's control.
To simplify things, じゃあ might be more usefully translated here as 'Well then', and さあ simply translated as 'Well...', as you wouldn't normally say 'Well then, I don't remember.'.
Not in that form. さあ can have many meanings.
At the beginning of a sentence (「さあ、。。。」) it usually means something like 'Well, ...'; 'Alright, ...'.
At the end of a sentence (often after te-form) it is similar to the sentence-ending particle よ and sounds assertive and masculine.
By itself, or with only a sentence-ending particle, it means 'I dunno.', or 'I have no idea.'. So, 「さあ。」or 「さあな。」would fit your example.
I believe English must be the unusual language here (and not Japanese) for this particular grammar rule, and now I can see why non-native speakers always trip up on English sentences like this. Indians for example will say "well, I am not remembering". Ask if they have something and they will likely answer along the lines of "yes, I am having a phone" instead of "yes I have a phone".
I wonder how many other languages this happens in? If anything, learning such a different language as Japanese, helps me to appreciate how difficult other people find learning English, and we should applaud others for their efforts to learn it.