NMR spectrum chemical shift confusion
It looks like you are describing continuous wave (CW) and not pulsed NMR. The latter is generally considered a more modern way of doing NMR as it was developed after the CW technique. The pulsed technique was also accompanied by the development of FT processing which resulted in a Nobel price being awarded to Richard Ernst.
In CW NMR you usually &quot;sweep&quot; the field - increase the external field $B_0$ in small increments, as you explain - while keeping constant the frequency generated by a &quot;read&quot; RF coil. Coincidence between the read frequency and the Larmor frequency $\omega = \gamma \left(1-\sigma \right)B_0$ - the resonance condition - results in a signal registered with a pickup coil. Here $\sigma$ is the chemical shielding which reduces the effective magnitude of the applied field $B_0$  at the nucleus.

by increasing the external magnetic field, won't we increase the energy differences between the spins and cause the resonance frequency to be higher which conflicts with the idea that we have a lower resonance frequency?

This is exactly the point of increasing the field. The Larmor frequency is related to the splitting in the energy levels, and you need a greater field to bring about a splitting in a more shielded nucleus, so that the resonance condition is satisfied.
The resonance frequency in such a CW experiment is always $\omega = -\gamma B_1$, but as the field at the shielded nucleus is $\left(1-\sigma \right)B_0$ and the shielding opposes the applied field, it is necessary to apply a larger field (&quot;go upfield&quot;) to excite a more shielded nucleus.