Dropping into BusyBox generally means the initramfs could not access the root filesystem. In other words, the root=/dev/sda1 part of your GRUB commands might not be correct, or your root filesystem might be damaged.
The GRUB commands you entered indicate that the second partition on your disk is probably the /boot filesystem, as the vmlinuz-<kernel version> and initrd.img-<kernel version> files are within the root directory of that partition.
Given that, the GRUB command set root=(hd0,1) is most likely not correct, but it might not be needed at all, so it probably won't do any harm. You could try using set root=(hd0,2) instead, for the sake of completeness, but it probably won't have any significant effect.
If your system boots in native UEFI-style and has Secure Boot enabled, then GRUB won't have to (and in fact cannot) load additional modules, as all the functionality will be built into the main grubx64.efi boot file. As your linux and initrd GRUB command lines have been specified with absolute GRUB-style pathnames, the set root= command should be completely unnecessary and could be omitted.
Since your disk seems to have GPT partitioning, it seems likely that your system might be using UEFI-style boot process. If that's true, then the first partition on your disk is likely to be the EFI System Partition (ESP for short), which would mean the Linux root filesystem would have to be the third partition.
GRUB will see all disks and SSDs as (hdN), whether they're SATA or NVMe. But the Linux kernel will normally see SATA storage devices as /dev/sdX (where X is a letter) and NVMe storage as /dev/nvmeNn1 (where N is a number). With/dev/sdX devices, the partition number comes immediately after the device letter; with NVMe devices, there is a letter p inserted after the whole-disk device name and before the partition number.
If you have a modern system (especially a laptop), your storage device is very likely a NVMe device.
Try changing the root=/dev/sda1 part of your GRUB commands to root=/dev/sda3 or root=/dev/nvme0n1p3.
If this allows you to get into the normal Linux environment, the easiest way to make the fix persistent is probably to reinstall GRUB with commands like:
sudo grub2-install /dev/sda
for SATA storage, or
sudo grub2-install /dev/nvme0n1
for NVMe storage respectively.
If this advice won't help, more information is needed: try running lsblk -o +fstype or cat /proc/partitions in the BusyBox prompt. These commands should reveal the sizes and ideally the filesystem types of each partition. If you can tell us that, more accurate advice can be provided.
The last 10 or so lines from the "lot of letters that fly by" might also include some useful error messages; please read them and copy any significant-looking error messages here.
grubin the first place? What had happened to require this?