In this paper, we extend recent approaches to Lexicalized Tree Adjoining Grammar (LTAG) parsing that combine supertagging with dependency parsing. In other words, we assign supertags (= unanchored elementary trees) to lexical items and we compute substitution/adjunction arcs between them. Kasai et al. (2017, 2018) jointly predict these structures with a neural graph-based parser. Predicting 1-best supertags and dependency arcs (as in Kasai et al. (2017, 2018)) however leads only to partial parsing due to incompatibilities between elementary trees and derivation trees. We therefore extend the approach described in Kasai et al. (2017, 2018) to n-best supertags and k-best dependency arcs and combine it with a subsequent A*-parsing step that extends the TAG parser from Waszczuk (2017). We show that this architecture allows for efficient full TAG parsing while being sufficiently accurate. We test our architecture on an LTAG extracted from the French Treebank (FTB).