Just tried to wear a tampon...

It sounds like you don't have it all the way in. If it is all the way in you soon won't even notice it, and it shouldn't feel like it's going to fall out, either. The only thing you need hanging out is the string.
Read your instructions again.

Is it inserted far enough?

Shove it further in - and remove the thing that you pushed it in with. Once it's in, you have no further use for the applicator.

As long as the string is outside your body, you have nothing to fear about getting it out again afterwards.

Try pushing it up higher. I never liked them though, can always feel them. I switched to a diva cup. Much more comfortable and can be left in for longer.