Has all changed, for the good too. 
If you go to the tourist info office in Torrevieja, walk past it (the church and square on your right hand side) and turn right at the third corner. About 100 yards up that road is the new office - they now have 8 desks all manned and computerised. You take a ticket and wait your turn - get there around 9.30am, no queues ! You need to show your Escritura, passports, NIE's and a utility bill. The lady prints and stamps your Padron forms.
Calle Union Musical Torrevejense 22. Its about 50 mtrs from the Palacio de Musica, and just before the OMIC office, it also has a long wheelchair ramp outside.