I went through this process in about 1986 when I had a fin keeler.
Try Chichester Harbour Conservancy 01243 512301
Hayling Yacht Co
Wilsons Boat Yard ( Hayling ) 02392 464869
Thornham Marina
And for Portsmouth try
Gosport Boatyard
Wicor Marine
I think you'll find moorings available alright, but check 4 important factors;
1, price,
2, Is there anywhere to store a rigid tender on shore - don't think about blowing up an inflatable each time, it wears out the inflatable boat and even a good one is likely to be too low freeboard, you'd be arriving with all your gear soaked.
3, is dinghy access restricted by tide ? It's often the case that the boat is happy floating in deep water, but you can't launch the dinghy to get to her !
4, How exposed is it ? Most people I know, and in my experience when I had deep water moorings in Chichester & Portsmouth, have substantial tenders to deal with the lumpy water in the exposed moorings positions.
I had a mooring at HISC for one season; it was way too exposed even for my large tube Zodiac 240 inflatable with 2hp engine, waves were coming over the side.
At that place I was let down badly by the advertised ferry launch deciding he couldn't be bothered, at Gosport boatyard there is a ferry launch too but in my experience reliable; it's too far out to the moorings to be realistic or safe for a normal tender, unless you get lucky with a closer mooring.
Bear in mind also, deep water moorings are exposed to thieves as well as weather, and insurers will only cover them for the summer.
In Chichester my first call would be to the Harbour Masters office.
You might be lucky .. sometimes the Nutbourne moorings are not all taken.
In fact according to the website there are 2 A3 moorings available (thats up to 9m loa) and an A2 suitable for a multihull ( 12m)
After the HM the next is to try the boat yards that have deep water moorings ( Thornham have some drying moorings but not deep water ones)
Just try all the yards and some of the chandleries ( eg the one at Hardway and Marine Warehouse at Fareham) as far as Portsmouth is concerned.
In some areas you could find yourself paying more for parking than for the mooring. It can be a particular problem when going away for a while rather than just day sailing.
That's where I am. The free car park is a couple hundred yards away and I have never seen it full. Peaceful area in the middle of a quiet new housing estate so feels safe to leave the car. I think if you want to stay longer than seven days the boatyard gives you a free permit. How long it will remain free for is another question.
The taxi service is very efficient and friendly though they do not carry a VHF nor mobile which is a bit annoying when you arrive back to the berth unexpectedly and need a pick up. Sometimes Sultans jetty is a bit shallow for me to pop by and arrange.
For 8m on a berth able to take 1.6m I pay about a grand a year which includes the ferry service.
The jetty has space to store your tender - you get a key for out of hours access,
And of course the boatyard itself is a good helpful friendly place!
I'd be very surprised if they don't have immediate space but it may be further into the harbour than you would wish.