Loose coolant, but no leaks

Depends on where it is leaking. if radiator or any of the rubber hoses, they have to be replaced. If it is the water pump or coolant pump, major repair to replace it.

Thermostat housing has O rings that can be replaced. Metal coolant pipes may get corroded at the tips that makes it loose its seal.

check the brake booster is working and not leaking. Also check the vacuum pump which may be incorporated into the alternator

When engine completely cools remove coolant from overflow reservoir to the cold mark on the reservoir which is the lowest mark on reservoir. If you continue to loose coolant you may have busted block.

flush the system and add new coolant as well as an additive

did you replace the thermostat when you put the radiator in,you could still have an air pocket in the system,recheck your coolant in the radiator when it is cold.
the next thing is to verify the cooling fans are working,when you see it hot check to see if it really is hot or gauge is lying,if it is hot see if the fans are running to maintain cooling temp,if they are not check coolant fan fuse and relay and last fan motors

Instead of having it replaced, you may be able to find a shop that will braze that area for you, which is an often-performed repair procedure on radiators.

At 34,000 miles, the coolant doesn't have to be replaced, but it won't hurt.

If it's dripping slowly when parked, it's dripping much faster when it's running. The coolant system is under pressure when the engine is running and up to operating temp, and that is probably causing coolant to spray out of that hole. You can check this by leaving the engine running and looking in the approximate location of the leak you found.

Continue to check and replace coolant/water as necessary until you get the problem properly assessed and repaired.

Head gasket by the sounds hence no leaks, the water it is getting burnt off in the combustion chambers.

You need a coolant pressure & test doing to confirm.

Make sure you keep your antifreeze level at 20° below zero. It is anti boil too.

The radiator is now plugged up, replace it before the overheating causes engine damage.

The time on yours more than miles affects the coolant, it should be changed every 5 years or 100,000 miles.