Few things to try, don't know if it helps, others will know better.
My GRUB Legacy format is simpler and works well, for example no { } or ' ' required:
title dCore-jessie
root (hd0,4)
kernel /boot/dCore/vmlinuz-jessie showapps tce=sda5 desktop=flwm
initrd /boot/dCore/dCore-jessie.gz
I don't think your GRUB configuration should need this line, root was already set with set root=(hd0,1) :
search --no-floppy --fs-uuid --set=root 81633858-9365-45c3-ba7b-09220bf2680e
Tiny Core automatically finds a dedicated swap partition, should be no need for a swap bootcode. There is a swapfile bootcode but it isn't relevant for you, as you want and have an existing swap partition. You could add the nozswap bootcode if you wish, less CPU overhead, as you plan to use swap partition anyway.
I don't think the 'linux...' line needs to specify yet again what root is. Also boot Tiny Core and run 'cat /etc/fstab', then try the device or mount point names instead.
I would use this, sub 'sda1' with whatever partition(s) you've set up for Tiny Core, sub desktop=flwm with whatever WM you installed:
menuentry "TinyCore"
set root=(hd0,1)
linux /tce/boot/vmlinuz home=sda1 opt=sda1 tce=sda1 nozswap desktop=flwm
initrd /tce/boot/core.gz