Hello fellow JRT owner! And welcome to Pet Talk!

Lexi owns me, she is in my siggy below!

If you really do not want to muzzle her, maybe you could get a Halti, its just like a horse halter, and you lead them where you want. It works very well and eventually she will just learn to walk where you want. It still gives her the freedom to pant, drink, bark, whatever, but if she goes near something you can just tug on the leash slightly and the little ring will move upwards on it and close her mouth, then say NO and lead her away.

-OR-

Take the training method, when your JRT goes up to something she wants to pick up something, tug the leash quickly away and give a stern and sharp NO! When she walks away when you say this, give her a treat and lots of praise! She will learn that picking things up does not get her a treat and more unsastisfied words. When she just looks at something or sniffs it, and DOES NOT pick it up, say GOOD GIRL! Give her a treat and walk away.

I hope that helps! Let us know how everything goes!