I've seen some people take on overgrown plots much later than now and be successful so it can be done with a lot of hard work but I appreciate the task you have on your hands with such a young family. However, I would agree with cooperman - cover what you can't manage this year, and doing small bits at a time rather than see a whole plot that needs to be cultivated in one go - I see too many newbies take on a plot and give up shortly afterwards because it all looks like too much work. I've had my big plot for 3 growing seasons of which I had covered 3/4 of it with black plastic sheeting or large sheets of cardboard, and dug beds as I went along and now have just one last bit to do this year
The squash bed was left covered and I cut holes through the plastic and filled the planting holes with manure/compost so I could still make use of the space that I didn't have time to dig over - something else you might want to consider in the short term
The bits you do clear, keep on top of the weeding from day one - the seeds brought to the surface during digging will appreciate the cleared ground and germinate as well as grow at an alarming rate, but if you catch them early, there are fewer later on in the year and of course they won't have grown to the flowering/seed stage to shed more seeds. Do be warned as you turn the soil over again for next year's season, some seeds can survive for many years so a fresh crop are exposed each time you dig over, not to mention windborne seeds like dandelion but you'll soon get the hang of it by then
Re: manure - it goes without saying that it does wonders for soil improvement and I'd say most crops do very well. Even carrots & parsnips ... but the higher nitrogen levels encourage their roots to fork (fang) so don't resemble perfect supermarket specimens and can be difficult to prepare for cooking if there are loads of roots which have intertwined which is one of the reasons you are advised not to dig in manure for these particular crops.