" The lowest are not on the new moon, but a few days after??"
That's correct Tengu. The easy ( if simplistic) way to explain that is although the height of a Spring tide is a result of the gravitational pull of Moon and Sun (Springs) there is also the rotation of the Earth to take into consideration which effects the massive movement of water on the Earth's surface. Imagine slowly swirling a big bowl of water around in your hands. After a few rotations you would see a sort of 'Tidal wave' running round the bowl. The Sun and Moon may be aligned and pulling together, but the spin of the Earth means the highest water takes a while (approx couple of days) to arrive.
Tides can be quite a complex subject so I'll shut up in a minute but one thing that you should be aware of for your own safety. Those published Tide Tables are..predictions..not gospel. To explain..
If a tide is shown in the table say as (example) 10 metres depth at high water in your area, but a weather depression with deep low pressure maybe 930mbs sweeps through, the tide height could be, in some areas up to a metre higher. (air pressure letting the water rise higher) Of course the same applies to high pressure, so your 'high' tide is not as high as predicted.
I can sometimes sail my boat with her 5 foot deep keel, back onto her river mooring up to 45 minutes later when the weather is poor (low pressure) than I can when Mr sunshine is beaming..

So, if you intend paddling about on the tides edge hunting Razors or whatever, just keep in mind that a rainy low pressure day...could..mean you'll have deeper water than you thought to wade through safely back to dry land. That's considering extremes mind you, usually an average weather low only alters things by a few centimetres.
Happy digging..