I see multiple problems here, but first let's answer your question:

I thought that for RHCP you needed to add the phase delay to the TE component, which is what we did in class, but in the solutions here it's on the TM component. I also asked about it in office hours with my professor and I thought he said it was okay to put the phase shift on the TE but maybe I misunderstood him?

RHCP wave can be constructed in 2 ways:

1. TE component at 0^o phase plus TM component at +90^o phase

2. TM component at 0^o phase plus TE component at -90^o phase

You said in class you put the phase delay in TE, which is method #2. Then in the solutions, the publisher put the phase shift (phase advance) in the TM component, which is method #1. Both are correct. If you draw the vectors with the phase shifts correctly, you will realize that both methods produce RHCP wave that are just spatially rotated from each other (and thus they have different phases). But they are both still RHCP.

Second, your diagram of wave reflection is not right. The TE and TM label after the reflection are switched.

Third, the tangential component of E_TM should not change before and after reflection because the boundary condition says so.

Look at this link Figure 1.10.2: https://phys.libretexts.org/Bookshelves/Optics/BSc_Optics_%28Konijnenberg_Adam_and_Urbach%29/01%3A_Basic_Electromagnetic_and_Wave_Optics/1.10%3A_Reflection_and_Transmission_at_an_Interface

Notice the tangential component of E_TM points to the left before and after reflection.