Paired Samples t test

I need your help to make sure i chose the right test for my thesis.
I want to see if children produce "Subjet-Verb order sentences" more than "Verb-Subject order sentences".
So ill compare for the same 20 children the means of their two scores (%).
Is paired sample t test the right test to choose ? knowing i have a normal distribution ?
Thanks in advance for your help.
Do you have the number of sentences looked at for each child
Each child has sample of spontaneous language of 30 minutes. Mean sentences per child: 342 sentences
Mean sentences including a subject-verb order: 5 ( Minimum: 0 ; Max: 27)
Mean sentences including a verb-subject order: 3 (Minimum: 0 ; Max: 10)
As the child isn't repeating the same sentence in the two forms, the content of one sentence does not influence the content of the other i.e. the each sentence is independent of the other. Furthermore you have in no way intervened to improve or correct the syntax of the sentence. So its not a befor after situation. So I feel its not a paired sample
I don’t think that a t-test would be correct.

It seems to me that each sentence can be a 0/1 trial with a probability p. So if each sentence is independent of the previous ones, then the sum of events would be binomial distributed with parameter n and p, where n is the number of sentences.

But it seems like the number of sentences varies wildly between children. I made up some data from the example:

Name SV VS n
John 30 70 5
Katy 40 60 20
Jane 50 50 7

Of course the different n will mean that the proportions p will be estimated with different accuracy.

It seems reasonable to assume that all children does not have the same population probability p, but that the probability of having a subject-verb sentence is different for different children. That would lead us to a mixed model where there is an individual random child effect and within the child there is a binomial variable about the probability of saying a subject-verb sentence.

A model like this:

log(p_i/(1-p_i)) = mu + b_i

Where p_i is the probability of child i of saying a subject-verb sentence, where mu is an overall estimate (for all children) of a subject-verb sentence, and b_i is an individual random effect (assumed to have a zero mean (which simply means that some children are above the over all average and some are below it).

This might sound complicated but it is just a usual mixed model that is available in most statistical packages.

I am curious if other think that this is a reasonable model?