It is trivial, changing the linestyle of the errorbars only require a simple .set_linestyle
call:
eb1=plt.errorbar(x, y, yerr=yerr, lw=2, errorevery=2, ls='-.')
eb1[-1][0].set_linestyle('--') #eb1[-1][0] is the LineCollection objects of the errorbar lines
eb2=plt.errorbar(x, y2, yerr=yerr2, lw=2, errorevery=3)
eb2[-1][0].set_linestyle('-.')