The very end of mRNA is polyadenylated. This is a fancy way of saying it ends on a lot of AAAAAAAAAAAAAAAAAAA. Even mRNA has had enough of 2020 it appears.
mRNA can be reused many times, but as this happens, it also loses some of the A’s at the end. Once the A’s run out, the mRNA is no longer functional and gets discarded. In this way, the ‘poly-A’ tail is protection from degradation.
Studies have been done to find out what the optimal number of A’s at the end is for mRNA vaccines. I read in the open literature that this peaked at 120 or so.
The BNT162b2 vaccine ends with:
UAGCAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAGCAUAU GACUAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAA
This is 30 A’s, then a “10 nucleotide linker” (GCAUAUGACU), followed by another 70 A’s.
The mRNA isn't intended to be copied in the body. Once the dose is used up there should be no more spike proteins created.
Here is a study of the effects - https://www.nejm.org/doi/full/10.1056/NEJMoa2024671.
It doesn't directly address your question, but it does show how the vaccine performs to control (labeled pbs) in terms of viral load and immune system activity when challenged with a dose of the virus.
The two shots have nothing to do with mRNA stability, they improve immunity, as in any other vaccine given in two or three shots.