We consider an intelligent agent seeking to obtain an item from one of several physical locations, where the cost to obtain the item at each location is stochastic. We study risk-aware stochastic physical search (RA-SPS), where both the cost to travel and the cost to obtain the item are taken from the same budget and where the objective is to maximize the probability of success while minimizing the required budget. This type of problem models many task-planning scenarios, such as space exploration, shopping, or surveillance. In these types of scenarios, the actual cost of completing an objective at a location may only be revealed when an agent physically arrives at the location, and the agent may need to use a single resource to both search for and acquire the item of interest. We present exact and heuristic algorithms for solving RA-SPS problems on complete metric graphs. We first formulate the problem as mixed integer linear programming problem. We then develop custom branch and bound algorithms that result in a dramatic reduction in computation time. Using these algorithms, we generate empirical insights into the hardness landscape of the RA-SPS problem and compare the performance of several heuristics.