Get a chromatic tuner. I use a Korg CA-30. It's not fancy, only cost me $25 new and I've had it for years.
There are loads of threads on how to properly detune a bass in the Hardware, Setup, & Repair section.
Most people, including me, will say dropping the tuning further than a full step should be done with a proper setup.
My advice for swapping between dropped D and C would be to go with a slightly larger gauge string, maybe a .115 or .120 low E string. There are many sources online for custom sets but a few manufacturers sell specialized drop tune sets you can try too. Set up action and intonation for whichever tuning you'll be playing the most, or better yet, have a properly setup bass for each tuning.
As stated earlier, attempting to play a drop tuned song with a lot of open low string pedal or galloped notes (mostly metal) on a standard tuned 5er is a pain. I drop the E to D on my 5 string if I'm gonna jam some Pantera or old Sepultura (Chaos A.D.).
I tuned my 5'r to CGC (keeping the 2 lower un-droped) that sounded better than tuing my 4'r to CADG or CGC(F?) but i'll keep the standar on my 4 and tune it down to D when i need to do that