There are two salient notes associated with the verb Ava-, and how we use it depends on how we interpret the dating of these notes.
First note, from War of the Jewels
"avo negative adverb with verbs, as avo garo! 'don't do it'; sometimes used as a prefix: avgaro. This could be personalized in the form avon 'I won't', avam 'we won't': ... no other parts of the verb survived in use, except the noun avad 'refusal, reluctance'.
This note is dated from between 1959-1960
Second note, from Parma Eldalamberon #17
"Sindarin (1) baw! imperious negative imperative "do not, don't!" avo, adv[erbial] negative with verbs in imperative, often as prefix, av, af, as avgaro, do not do it! Also personalized avon, I won't, avam, we won' t. Cf. avad 'refusal'...
"Another group of etymologies comprises most of the content of a dozen loose sheets place[d] together, the first of which is dated "Dec. 59 ... The second group of sheets begins with a three-page discussion of the expressions of the negative in Elvish, with the heading "Definitive Linguistic notes" (see ..JABA, BA) ... One of the half-sheets has on the back a partial typescript page of a draft of "Quendi and Eldar," which suggests a date around 1959 or 1960 for these documents."
Unfortunately this dating of 1959 to 1960 for the second note makes the two contemporaneous, leaving us at a quandary of which one was written last and thus which one supersedes which.
Differences between the two notes
The first note states that Av- was 'sometimes' used as a prefix. The second note states 'often as prefix'. While the first note explicitly states that there are no other surviving parts of the verb ava- apart from Avon = I won't and Avam = We won't, the second note does not make this distinction. It does however follow directly on from the Quenya note of the same (my bold) - "ava/va was often personalized in Quenya in 1 st person singular, 1 st person plural exclusive only: vanlvanye and vamme, I won't, we will not" While it is possible that the Sindarin note excludes this quantifier because it only applies in Sindarin, I think it more likely that Tolkien intended the Sindarin note to have the same meaning as the Quenya. (He often wrote a fuller note in Quenya and then followed it with 'in S. ...')