I live about 20 minutes drive from Crawley and go to Gatwick quite a bit for spotting but have not been to Crawley really too much. Sorry. A few things you could do though:
Spotting at Gatwick. Good stuff on 26L.
Drink at the Hotel Ibis. They have a couple of nice benches outside and sometimes the occasional ramper is out there.
Drink at the Flight Tavern, just adjacent to runway 26 but no good runway views. Nice though.
Go to Redhill shopping.
Go to Redhill aerodrome. Cubair is a nice school and a cafe too.
Go to Kenley aerodrome near Caterham. Surrey Hills gliding club are there weekdays depending on weather. They are very nice guys and welcome you just to watch or will take you up if you have a bit of cash and like to.
Go to Denbies vinyard outside Dorking. Beautiful setting.
Walk a couple of days of the North Downs Way. There are guidebooks online and in libraries and you can print out the OS
map sections you need for free online.
Go to Dublin for the day with Ryanair (if you don't mind them). Cost is cheap if you book well in advance and they go at least 4 times a day.
Go to Europe for the day with Easyjet. So many options. Toulouse airport is nice and there's an A380 factory tour if you can take a taxi.
Go on a day trip to Calais or even just Dover to watch ferries. Bruges in Belgium and Lille in France are both easily reachable in a day if you have a car and are both nice.
Try paragliding with the Green Dragons club (if you like adrenaline!) They are also nice guys. They are very near Kenley about 10-20 minute drive from Crawley.
Golf. There is a nice public club at Rusper (very rural with good aircraft spotting while walking around if the wind is good). There's also a nice 9-hole small course just near the airport where you can walk round and watch aircraft landing 26L.
Don't know if any of those will be attractive to you but thought I'd write anyway.
Good luck with the job and the relocation!