something like:

find dir1 dir2 -type f -name "*.fa" |\
   awk '{printf("mv \"%s\" \"%s\"\n",$0,$0);}' |\
   sed 's/\/\([^\/\.]*\)\.[^\/]*$/\/\1\.fa"/' > check_this_script_then_execute.sh

GNU parallel is also an option.

parallel -j1 --dry-run mv {} {= s/\\..+/.fa/ =} ::: *.fa

Remove --dry-run if the commands look good.

You may need to modify the directory structure or regex depending on your actual directory structure and file naming.

Also try brename. It's handly and safe, being able to check potential conflicts and errors:

Removing anything after the first dot (-p '\..+' -r "") but without changing the file extension (-e). -R is for rename recursively (including sub directories), and -d is for dry run.

$ brename -R -e -p '\..+' -d
[INFO] search paths: ./
[INFO] 
[INFO] checking: [ ok ] 'GCF_029211165.1_ASM2921116v1_genomic.fa' -> 'GCF_029211165.fa'
[INFO] 1 path(s) to be renamed

Usually, I kept the versions (.1).

$ brename -R -p '^(\w{3}_\d{9}\.\d+).+' -r '$1.fa' -d
[INFO] search paths: ./
[INFO] 
[INFO] checking: [ ok ] 'GCF_029211165.1_ASM2921116v1_genomic.fa' -> 'GCF_029211165.1.fa'
[INFO] 1 path(s) to be renamed