We investigate the global texture model of structure formation in cosmogonies with a nonzero cosmological constant for different values of the Hubble parameter. We find that the absence of significant acoustic peaks and little power on large scales are robust predictions of these models. However, from a careful comparison with data we conclude that at present we cannot safely reject the model on the grounds of present CMB data. Exclusion by means of galaxy correlation data requires assumptions on biasing and statistics. New, very stringent constraints come from peculiar velocities. Investigating the large-N limit, we argue that our main conclusions apply to all global O(N) models of structure formation.