I have done that trip in the other direction, but you are not allowed to cross the border by foot. I cannot comment on entering Kaliningrad Oblast from Poland by foot, as I have taken the bus from Gdansk. It seemed more difficult to enter Russian territory than to leave it in a slightly unusual way. Nick's tip on an alternative border point might be worthy checking out.
Exploring the stretch of the Curonian spit between Zelenogradsk and Nida is something I can highly recommend. While the coast of Kaliningrad oblast up to and including Zelenogradsk is extremely touristic with a negative effect on nature, the parts further back on the Curonian spit are beautiful and calm.
The border crossing at Morskoe is very quiet. There is a bus that you can catch there, however it doesn't go very frequently and you might have to make a deal with the driver to drop you off earlier behind the border crossing. In my case I hitched a ride over the border after about 45 minutes of waiting time. I got help from the border guard who asked locals crossing the border in their car to pick me up.
Another thing to consider is that you officially need to register your place of residence if you plan on staying for longer that 10 days. For this it might be easier to stay at a hotel for at least one night in Kaliningrad and acquire the invitation for a visa through them. My couchsurfing host was not able to register me, it was refused on the basis that my invitation was not from him, so I had to rush my trip a bit. You might get away with a fine when exiting the country if you stay longer.
I understand that this was a bit more descriptive than a straight up answer, but I hope it helps with your trip and good luck!