I want to know if it will be okay or if I should shovel out the water now that it has completely flooded.
Does this have a long-term solution as well?
Get that excess water out of those pots ASAP before the plants drown.
Pots without drainage holes… amateur move.
Yeah, I’ve had pots with drainage holes flood too—sometimes the rain is just relentless and overwhelms them. Just saying.
Same. My poor chives 
Try adding coarse sand or pumice to prevent this. Rain usually has plenty of oxygen, so if your plants drowned, it was likely due to a hypoxic soil mix when kept wet too long. Based on experience from growing cuttings in water.