Agree that astroturf will likely make the problem worse, even with drainage.
Add as much drainage as you can - like one poster said above, you can read up, watch YouTube and DIY to save on costs (and will probably do a better job!).
Add a good mix of shrubs to improve drainage, dig really deep when planting and add lots of gravel before they go in. Their roots will take up water in winter, although bear in mind most go dormant so there's always a couple of months where plants/trees are taking up less water.
Add some paths - again you can do this yourself - in areas where footfall is heavy. This will reduce people walking on other parts of the lawn, in turn reducing the chance of soil being compacted in those areas.
Aerate the soil every autumn, you can do this by walking around with spiked shoes! Basically, making little holes all over the surface of the lawn to aid drainage / stop the ground getting so compacted.
Try growing a mixed lawn: encourage moss, clover, wildflowers and other non-grassy plants (that you can still mow), as the mix of plants stops grass forming big, impenetrable 'mats' of roots underneath the surface, which isn't great for drainage. It's harder to do on clay soil but any diversity in your lawn is a good thing. Again, can be done entirely DIY (as I did mine this way!). Lots of free advice and videos on how to do this (can be done in small sections, you don't have to dig up your whole lawn in one go).
Consider growing seasonal flowers in the lawn itself, like snowdrops, primrose, daffs, crocus etc. They tolerate wet, shady lawns and more importantly grow in winter - so will take up some water, and provide a bit of colour. If you leave their leaves to die back in spring you can mow as normal the rest of the year.
Growing them either side of your new 'paths' also means people will more likely stick to the paths in winter months and not trample/compact the soil, so it's a win-win.
Good luck - I had a shady, boggy garden for years so feel your pain :)