if I remember correctly, mutations happen during transcription
They certainly can, and there is some evidence that highly transcribed regions are more prone to mutation, but that is important principally in somatic mutations, in the germ line mutations which actually contribute to heritable variation this effect will be considerably reduced. It is by no means that case that mutations only happen during transcription, perhaps more important in evolutionary terms are meiosis and mitosis which similarly involves the unwinding of DNA to allow DNA synthesis enzymes to have access to single stranded DNA. You are basing your reasoning on a number of huge assumptions.
Thus these mutations become fixed in a population through genetic drift, which is a random process. This is without counting that, from an evolutionary point of view, the majority of the genome is composed of junk DNA, and so mutations inside this DNA are not affected whatsoever by NS.
Wow, that is the sort of argument that creationists always use as a strawman of what biology says. I don't think anyone familiar with modern biology really believes this. The majority of the genome is composed of non-coding DNA, but this isn't the same thing.
Unless you specifically define 'Junk DNA' as DNA which is neutral in terms of selection, i.e. non-functional, then any of the commonly used meaning of the term show your statement to be wrong. There are a number of non-coding sequences in the genome that have been targets of selection. Many of these serve structural roles in stabilising the chromosome, see microsatellite DNA, Telomeres, Satellite DNA and probably othe as yet uncharacterised sets of sequences. There are also regulatory elements which are found to be embedde in intergenic regions which might previously have been discounted as 'Junk'.
TTFN,
WK