Greys GRX-S Bass
It's right at the upper limit of your budget though.
They do an 11' 9" and a 12' 6" version.
I have the Shakespeare Salt Extreme Bass and the 12' 6" Greys GRX-S Bass and the Greys makes the Shakespeare look like a toy by comparison.
That's not to say the the Shakespeare is a bad rod - it isn't. It's a lot of fun to use and very capable, and if you plumped for one you'd have enough left in the budget for a reel to go with it.
The Shakespeare is a 2 - 4oz rating and I find it will happily throw a 3oz sinker and a big bait a very long a way. I use an Abu 6500 CS CT Mag high Speed multipler on mine with 18lb mainline and it can really throw and is super-light and comfortable to hold for long periods.
The Greys is a 3 - 5oz rating, making it a far more versatile rod in my opinion or, at least, it's more versatile to the kinds of fishing that I do, which may not match the fishing you will do.
My plan is to use these rods almost exclusively until Autumn when the bigger rods will come back out to deal with the heavier seas.
Don't forget that Shakespeare also do a 'flattie' version of the Salt Extreme, which has a lower 1 - 3oz rating and a much tippier action. I only mention it because this is a factor you specified.
Both of the above rods fall under your 13' spec but they are both great all rounders for a variety of sea angling conditions around our coast. As a coarse angler you will probably already have a good idea of the various 12 and 13' carp and pike rods that could also be a suitable solution for you.
Where are you based ?
I am just north of Whitley Bay and I am out fishing often - if you were close enough to arrange a meeting you could see both of the above rods side-by-side and have a throw or two with them if you like. Waving a rod around in a shop gives you little idea of its potential. Both the Salt and the Greys feel positively floppy until you put a reel and sinker on when, all of a sudden, they start acting the part.
Good luck whatever you decide on
