Last year I felt overfaced by a whole plot and as you pointed out, the weeds become overwhelming. I spent much of the winter months setting up raised beds and I can honestly say it is a much better system and so much more manageable. Planning crop rotation is so much easier too. The "no dig" policy is only partially successful because despite digging deep to clear the bindweed, it still keeps coming up, so I will have to have another go this autumn/winter!
One thing I thought worked brilliantly - during the spring months when my lawn need mowing, I dug in the grass clippings into a bed, mixed with rotted manure, then covered the single beds with thick black plastic (even using the empty compost bags were useful) It warmed up the soil (being black), the worms and bacteria got to work and broke down the grass, weed seeds germinated under the plastic then withered and after a month, the bed was ready for planting. And guess what, there were even fewer weeds to have to pull up later on!
The one problem is getting hold of wood. Half my plot consists of wood I scavenged, but in the end I had to buy some and that got a bit expensive. I'm hoping the investment will pay off in the long run.
As for using polythene on the footpaths, please be careful - when wet, it can be very slippy and accidents happen when you are physically tired. Try laying bark or wood chippings over (stone chippings are better but expensive), or use a material membrane instead.
PS forgot to mention that I have now got loads of ants nests that have taken up residence where the wood supports are!