The early PS times probably are booked up. Normally, LTT does have a 60 day booking window. However, for
MNSSHP (and
MVMCP), Disney is/was offering a "add on dinner package" at LTT available to those who purchased MNSSHP tickets. Essentially, all the dinner package consisted of, was a PS for the LTT that you didn't have to wait until the 60 day window to book. Many people did take advantage of it. You still pay the same price and you don't have to pay until after your meal.
Yes, if you get a reservation at 6 p.m., you can stay in the park between the time the MK closes and the time it reopens for the party.
Yes, the dinner will still be a character dinner.
If you can't get a good time for a PS by calling now, try again at the 60 day (or maybe even the 59 day) mark. Some people will cancel their LTT PS if they get a PS at one fo the other sit down restaurants - Tony's, the Plaza or CRT. If you get a late dinner ressie at one of those (something from 5:15 on), you'll also very likely be able to stay in the park between closing and reopening. These restaurants will not be open during the party, though and IIRC, the latest PS you can get at one of them would be 5:50 p.m. Above is a description of how it worked in the past, so I assume it will work the same way this year. Can't be sure, though.
HTH.