We never had an allotment, but we did have a huge (2 acre) garden where we had set aside various areas for veg, fruit, fruit trees, herb and a greenhouse. My in-laws have had an allotment the entire time I have known them - around 20 years now.
You will find that allotments vary from one place to another - looking for a secure allotment if you live in a city is useful. My in-laws have this issue and it is not uncommon for sheds to be set on fire and gardening equipment stolen (prior to the site being made more secure that is).
On the allotment itself you will get to know people and if you are willing to 'help' them they will help you and if you are willing to spend time listening to people, you will learn a lot. Time is the key though and from your post I am not certain you will have enough of it, without knowing the ages of the 4 children you have. My in-laws also have 4 sons and all 4 were 'made' to help out on the allotment. All but one of them sees it as a chore now!
You will also find that the allotment probably has a gardening club where seeds can be purchased much cheaper than off the shelf in supermarkets or garden centres. This will help considerably because the costs do build up - something I could not share with others having no neighbours where we lived at the time.
As for seasonal variation and surplus veg. Well season is inevitable, but there are crops that are suitable to all year round harvesting and sowing and some actually require it (such as onions, garlic, kale, broccoli). I always grew around 50kg of many types of potatoes each year, and we would not need to purchase potatoes again until around Easter - but they did need checking over regularly to remove any bad ones and eat the ones showing signs of 'going soft' first. They also needed to have their skins dried on them, so a day out in the sun was required to harden the skins after I had washed them (I established early on that cleaning the skins before the soil had dried was much easier than several months down the line).
Some veg, mostly the leafy kind can be grown on a rota, so you sew a few seeds each week of say carrots, beetroot (for the leaves and later for the root itself), peas, broad beans, runner beans, any salad leaves etc, so that you can harvest regularly and have a regular crop.
But things like carrots, beetroot leaves, lettuce and to be honest, even runner beans and broad beans can be grown in any garden border, as can herbs, so perhaps your best option would be to consider growing a few things at home this year and see how it works out?