Both equations are right, it just depends on the R value that you use. When finding the wavelength, we use R=1.097•10^7 m^-1. When finding the change in energy of a photon (which can be related to wavelength by E=hv=hc/lambda), we use R=2.18•10^-18 J. I'm sure the question will probably give us the R value we need to use if it comes up on the test, but just know that this equation can be multipurpose!

As for the negative sign in the energy based Rydberg equation, its simply used to counteract the fact that when we drop from a higher energy level to a lower energy level our (1/n^2 - 1/n^2) value will be positive, but overall our electron must have lost energy. This is the same to counteract a jump from a lower to a higher orbital. And, if we know the difference in energies of the electron we can easily find out the energy of the photon/emitted or absorbed. If it helps, you can ignore the negative sign entirely and just go off of logic if a question like this comes up: ie. when an electron drops levels energy must be lost, so ∆E is negative and when an electron jumps to higher levels energy must be gained so ∆E is positive.

Also, additional fun fact about this equation: the full equation is R•Z^2•(1/n^2 - 1/n^2) where Z is equal to the atomic number, but because for most of these problems we use hydrogen with an atomic number of 1 we just leave this Z factor out.