Continuously tunable random laser basing on rhodamine 6G was achieved. Wavelength tuning was implemented by wedge-shaped sample shift relatively to the pump beam. The physical base of the tuning method is gain band shift caused by reabsorption. We ascertained that two medium thickness ranges of significantly different tune efficiency exist. The lasing characteristics of the laser were investigated depending on active medium and pump beam parameters. The feedback formation mechanisms in the random lasers of different thickness were analyzed.