Response-adaptive (RA) allocation designs skew the allocation of incoming subjects toward the better performing treatment group based on the previously accrued responses, while covariate-adjusted response-adaptive (CARA) designs additionally condition allocation on a set of patient characteristics. Because unstable estimators and increased variability can adversely affect adaptation in early trial stages, Bayesian methods can be implemented with decreasingly informative priors (DIP) to overcome these difficulties. DIPs have been previously used for binary outcomes to constrain adaptation early in the trial, yet gradually increase adaptation as subjects accrue. We extend the DIP approach to both RA and CARA designs for continuous outcomes. Simulated clinical trials comparing the behavior of these approaches with traditional RA, CARA and balanced designs show that the natural lead-in approaches maintain improved treatment with lower variability and greater power.