Be careful not to drown your plants. Try to match the amount of water you give them to the amount of water they need and are using. Allow the medium to dry until it is just moist before watering them again. This is especially important for newly planted or transplanted VFTs, which are already dealing with some shock to begin with.
Although they need a little boost of a few more waterings to begin with, until their roots become established, they still need to dry out quite a lot between waterings, with plenty of moving fresh air and air entering the soil.
If you give plants too much water (this is true for many plants) they will often respond by growing faster, but that growth can be very weak, unheathy, and prone to rot or early death.
It's like the plants are being forced to drink too much and have to try to grow faster (which can be stressful) to use the extra water in order to avoid rotting and the fungal infections that can be caused by constantly sitting in soggy, waterlogged conditions. Venus Flytraps are not aquatic. They are moist savannah plants, not swamp plants.
These are just my opinions and I'm sure there will be a variety of probably conflicting opinions, but you may wish to consider what I have just mentioned.
VFTs can tolerate sitting in standing water or even being submerged for a few days, but it is not healthy in the long run, in my opinion. The only time I ever leave pots of VFTs in trays of water (if I'm home to attend them and not on a vacation), is sometimes if they are outside on a hot day, in full sun, with plenty of fresh air. In shadier or cooler conditions or where the air is not moving as much, I never (well, rarely) leave them for long in water. It's too stressful and risky regarding microbial infections and rots of various sorts, and they grow healthier with less water and more air around their roots, in soil that is moist, not soaked.
Be happy in the travel--there is no destination