There are many ways to eat in Palanga, from smoked fish bought at the market and eaten on the beach to a fish restaurant where a table is worth booking. The trouble is that in July everything on J. Basanavičius Street looks the same, and the price does not necessarily tell you where the food will be good.
Below are the addresses worth knowing, and a few notes on seasonality that will save you time. Opening hours in the resort change all the time, so check before you set off.

The main pedestrian street runs for about a kilometre from the town centre to the pier. It is certainly touristy, but it has a few places that have been open for years:
At the end of the street, at the junction with Love Avenue (J. Basanavičiaus g. 45), stands Šachmatinė, a modernist pavilion that has long been a Palanga landmark. Today it houses a pizzeria with a terrace; even if you do not eat there, the building itself is worth seeing.
Closest to the hotel, in the building next door at S. Dariaus ir S. Girėno g. 13, is RestoBaras Kupeta, a restaurant and café that serves food and also hosts concerts and stand-up nights. Its published opening hours are 10.00–22.00.
A little further on, at Birutės al. 47, is Vandenis, a restaurant with a music club that is also open in the evenings.
Vytautas Street is the second place worth looking. It runs parallel to the seafront, with fewer passers-by and more places where Palanga locals eat, not only holidaymakers.
If you want more options, the most complete and reliable list is on the Palanga Tourism Information Centre website, visit-palanga.lt: it lists the resort restaurants that are actually open, not advertisements.
Palanga market (Senojo Turgaus g. 12) is a place many holidaymakers never reach, although it is only a few minutes from Vytautas Street. The food hall sells fresh and smoked fish.
The market is open Tuesday to Friday 8.00–17.00 and Saturday 7.00–16.00. The choice is wider in the morning, and fewer traders remain after lunch.
Near Palanga, in Kunigiškiai (Žilvyčio g. 1), is the Pas Berniuką smokehouse, one of the addresses where fish is bought straight from the smokehouse rather than from a stall.
The best summer dinner in Palanga is often the one nobody cooks: smoked fish, bread from the bakery, something from the market and the beach beyond Love Avenue. From the hotel to the sea is about five minutes, so the food does not even have time to cool.
Two rules are worth knowing. First, walking and sitting on the protected dunes is not allowed, so have your picnic on the beach, not on the slope of a dune. Second, bins are not always to hand on the seafront, so it is best to bring your own bag.
In September and October the same plan works even better: the beach is empty, and smoked fish eaten in warm clothes is perhaps at its tastiest.
One more note that will save you a kilometre's walk: the Kurhaus (Grafų Tiškevičių al. 1) is not a restaurant, although the name sounds like one. Today it is the Palanga Culture and Youth Centre, a venue for concerts and plays.
From June to August you are unlikely to get an evening table in the more popular restaurants without a booking, especially at weekends. It is worth calling at least on the morning of the same day.
Off-season the picture is the opposite: seasonal kiosks and outdoor terraces close, and it is mostly the restaurants and bakeries in the centre, by the pier and in hotels that stay open. In autumn and winter opening hours get shorter, so it is best to call before heading to a specific address.
One more summer detail: on J. Basanavičius Street, prices rise and queues grow the closer you get to the pier. Go a block or two to the side, to Vytautas Street or towards the park, and you will often get the same thing for less and without the wait.