Oversampled adaptive sensing (OAS) is a recently proposed Bayesian framework which sequentially adapts the sensing basis. In OAS, estimation quality is, in each step, measured by conditional mean squared errors (MSEs), and the basis for the next sensing step is adapted accordingly. For given average sensing time, OAS reduces the MSE compared to non-adaptive schemes, when the signal is sparse. This paper studies the asymptotic performance of Bayesian OAS, for unitarily invariant random projections. For sparse signals, it is shown that OAS with Bayesian recovery and hard adaptation significantly outperforms the minimum MSE bound for non-adaptive sensing. To address implementational aspects, two computationally tractable algorithms are proposed, and their performances are compared against the state-of-the-art non-adaptive algorithms via numerical simulations. Investigations depict that these low-complexity OAS algorithms, despite their suboptimality, outperform well-known non-adaptive schemes for sparse recovery, such as LASSO, with rather small oversampling factors. This gain grows, as the compression rate increases.