I am not sure but if it is the case, anything like veet or nair would cause the same problem.
I would just trim them short rather than shave and wait until you can wax again.
Have you tried wax for short hairs (cream wax is a good option). Or hot film wax. Both work well on short hairs. But you will need a wax heather and the waxes don't seem cheap but the tubs usually last a while.
for reduction on Underarms, Bikini, Full Legs & 3/4 Arms. Skin type IV
Electrolysis - Further details in: My sister's electrolysis diary
[27hrs of Blend, April 2008-Dec 2010 in UK, for coarse hair on lower sideburns, coarse chin hair, completed upper lip, shaped eyebrows]
[Sept 2011 to Nov 2013, once yearly sessions with Josefa. Completed reduction of facial/neck fuzz in approx 27 hrs TTT]