What is a simple formula to approximate the change in relative humidity as temperature rises/falls?
If you want to convert the %RH from one temperature to another you can use one of various parameterizations, for instance the Antoine equation:


  range: 273. - 303 K,
  Bridgeman and Aldrich, 1964,
  https://webbook.nist.gov/cgi/inchi?ID=C7732185&amp;Mask=4#Thermo-Phase
  
  A = 5.40221;
  B = -1838.675;
  C =   -31.737;
  
  log(P) = A + B/(C+T);


Then obtaining the $RH_2$ at $T_2$ from that at $T_1$ is as simple as
$RH_2 = RH_1 P_1/P_2$, where $P_1$ and $P_2$ are the saturation vapor pressures of water computed with the equation above, assuming the ambient pressure (and therefore the partial pressure of water) is constant.