We report on measurements of electron transport for the fractional quantum Hall effect (FQHE) at filling factors nu = 2/3 and 3/5, in magnetic fields B(t) tilted be angles theta with respect to the normal to the sample plane. Our device was prepared at an electron density of only 2.4 x 10(10) cm-2, but still exhibited a well-developed FQHE at nu = 2/3 and 3/5. This exceptionally low density allowed us to access very low total fields, where the spin is less likely to be completely polarized. For many tilt angles, we obtained gap energies DELTA from the temperature dependence of the diagonal conductivity on the FQHE minima. For both 2/3 and 3/5, plots of DELTA versus B(t) exhibit minima that are accompanied in transport by splitting of the FQHE. For 2/3 the minimum in DELTA(B(t)) is sharp and deep, with DELTA reduced by 70%. With B(t) well above its value at the minimum, DELTA(B(t)) for nu = 2/3 is linear, with slope almost-equal-to g-mu(B) for GaAs, indicating an increase in the two-dimensional electron-system Zeeman energy on excitation. We present a detailed survey of the evolution of the splitting of the FQHE with angle, and find that local rho(xx) minima that are shifted up to 6% upfield of nu = 2/3 at theta almost-equal-to 23-degrees evolve continuously into an unsplit FQHE at nu = 2/3 at theta almost-equal-to 0-degrees. The split and shifted FQHE's that we observe are interpreted as effects of phase separation associated with ground-state spin transitions.