I would additionally advise you to go to a "Dollar Store", and buy several $1 toothbrushes for scrubbing the old finish & stripper out of the checkering, then tossing it to use another for staining and yet another for brushing in some finish, after the rest of the stock is finished.
(I mask off the checkering after stripping, and stain/finish it separately/last.)
While I too use CitriStrip, I only use it for epoxy-type finishes.
For most others, I first try/use Formby's Furniture Refinisher (following the can directions), also available @ Home Cheapo, etc, to remove the old finishes.
I would let any stripper dry a looong time ( I do overnite), to ensure all chems have fully evaporated.
FWIW, except for repairs and/or repaired areas, I haven't had to touch a stock with sandpaper (or de-whisker one) ever since these stripping products came on the market.