I've learnt to never cut my lawn when it's:
Damp, wet.
During dry, hot spells (generally July & August here in Missouri), when temperatures are constantly over 90°F (32°C).
Yes, my grass does on occasion get somewhat long, but never long enough for it to become an eyesore -- at least I don't believe so (and I'm not living in denial here, honest), nor have I had anyone complain to me directly or indirectly about it...
As you've discovered, cutting grass when it's damp (wet) is difficult, plus:
You won't get a good cut. Period.
The grass will mat-down, smother itself.
You increase the possibility of fungal diseases taking hold in your lawn -- cut, wet blades are a lot more susceptible to allowing those kind of diseases to enter...
You (greatly) increase the possibility of thatch buildup in your lawn, which it turn can lead to fungal diseases taking hold...
If you must mow your lawn when it's damp, wet, it is highly recommended you collect up the clippings as you go (via a bag attached to your mower). Why?
Even a mulching type mower will not breakdown the clippings so they "disappear" into your lawn.
The wet clippings will instead somewhat help mat-down your lawn and will most definitely add material to your lawns thatch layer.
If you wish to read about my "organic" approach to lawn care, go here on SE.