Check to easy things first:
Battery Connections, Clean and Tight(wrench, not screw driver)
Ground Connections, same as above.
Check ECM wires while at battery, green and black known to break
Check Vacuum lines for cracks
Could be Petcock going bad. Check it by applying vacuum and make sure it flows fuel freely.
Yes could be rust or junk in tank filter
Could be pinched line at right rear of tank -remove gas cap and listen for sucking sound.
2005 VTX-1300R PAIR done, Stock FURY Pipes, Decapped Airbox, K&N Filter, #210 Main Jet, #58 Pilot Scar Mod with 1 shims, AF @1.875 turns. 10 Min AF Adjuster, All Balls, Nylon Shock Bushings, 1800F Risers, Avon Front w/Dyna Beads 205/70/15 Grabber UHP rear w/Airsoft Pellets, J&P $40 Passenger Floor Boards, LED Red Run/Stop Lights in Yellow lens, Rotella T6 Synthetic 5-40w
Ultimate Seats! Headlight Relay