A novel fully-connected (FC) tri-hybrid beamforming (THB) architecture is proposed for pinching antenna systems (PASS). In contrast to conventional sub-connected (SC) PASS, the proposed FC architecture employs a tunable phase-shifter network to interconnect all radio frequency (RF) chains with all waveguides. This facilitates a THB framework that integrates conventional hybrid analog-digital beamforming with pinching beamforming. A weighted sum-rate (WSR) optimization problem is then formulated to jointly optimize the transmit beamformers and pinching antenna (PA) positions. Two algorithms are developed to address this challenging non-convex problem. 1) Fractional programming (FP)-based algorithm: This algorithm directly maximizes the WSR using an FP-based alternating optimization framework. Particularly, a success-history based adaptive differential evolution (SHADE) method is proposed to optimize PA positions, effectively addressing the intractable multimodal objective function. 2) Zero-forcing (ZF)-based algorithm: To reduce design complexity, zero-forcing is employed for transmit beamforming. The PA positions are subsequently optimized to maximize the WSR via a modified SHADE method. Simulation results validate the effectiveness of the proposed algorithms, revealing that the FC-THB PASS achieves WSR comparable to the SC architecture while delivering superior energy efficiency with fewer RF chains.