The TRAMP complex contains two enzymatic activities essential for RNA processing upstream of the nuclear exosome. Within TRAMP, RNA is 3' polyadenylated by a sub-complex of Trf4/5 and Air1/2 and unwound 3' to 5' by Mtr4, a DExH helicase. The molecular mechanisms of TRAMP assembly and RNA shuffling between the two TRAMP catalytic sites are poorly understood. Here, we report solution hydrogen-deuterium exchange data with thermodynamic and functional assays to uncover these mechanisms for yeast TRAMP with Trf4 and Air2 homologs. We show that TRAMP assembly constrains RNA-recognition motifs that are peripheral to catalytic sites. These include the Mtr4 Arch and Air2 zinc knuckles 1, 2, and 3. While the Air2 Arch-interacting motif likely constrains the Mtr4 Arch via transient interactions, these do not fully account for the importance of the Mtr4 Arch in TRAMP assembly. We further show that tRNA binding by single active-site subunits, Mtr4 and Trf4-Air2, differs from the double active-site TRAMP. TRAMP has reduced tRNA binding on the Mtr4 Fist and RecA2 domains, offset by increased tRNA binding on Air2 zinc knuckles 2 and 3. Competition between these RNA-binding sites may drive tRNA transfer between TRAMP subunits. We identify dynamic changes upon TRAMP assembly and RNA-recognition motifs that transfer RNA between TRAMP catalytic sites.