Yes. Devotees from other states can book Darshan and Sevas at Srisailam. You do not need to be a resident of Andhra Pradesh.
This applies to Sparsha Darshan, other Darshan services, and Sevas that are available for booking.
When booking through WhatsApp Mana Mitra, the District selection refers to the district where Srisailam Temple is located, not the devotee's home district.
For Srisailam Temple, select Nandyal as the district and then select Sri Bhramaramba Mallikarjuna Swamy Temple.
So, if your own state or district is not shown in the booking options, that does not mean you cannot book. Select the temple's district, Nandyal, and continue with the booking.
In short: Devotees from any state can book available Darshan and Seva tickets at Srisailam. The district shown during Mana Mitra booking is the temple's district, not the devotee's residential district.