Licia of Lindeldt will not relocate

There is a merchant near a bonfire in the Forest of Giants who will sell lifegems in limited supply. After talking to her a few times, and defeating the boss of the forest, she'll move to Majula, at which point her supply of lifegems will become infinite. You'll know you've talked to her enough because she'll talk about it being time for her to move on.

However, it seems your real problem is that you need more Estus Flasks. You're able to get more by finding Estus Flask Shards and giving them to the Emerald Herald in Majula. There are three Estus Shards you can find fairly early on.

The first can be found inside the well by the mansion in Majula. Just knock over the rock down the well and an Estus Flask Shard will appear.

A second one can be find in a room above the second bonfire in the Forest of Fallen Giants. The door will say it's locked, but you can actually break it open if you just attack it.

Finally, you can find another Estus Flask Shard inside the mansion. You'll get the key from Cale the Cartographer inside the Forest of Fallen Giants. You'll find him shortly after the second bonfire inside the Forest of Fallen Giants. Just go down the ladder next to the bonfire, and when you get to the area with the large tree drop down to the area below. You'll then find him inside of a cave on the side opposite where you came in. Just keep talking to him and he'll eventually give you the key to his mansion.

The only places where you can find Licia of Lindeldt are in Heide's Tower of Flames or in the way between Heide's Tower of Flame and Majula, the only way known for her not to be there is after she's killed (Either you invade her with the Crushed Eye Orb or simply kill her). After she's killed a tombstone MIGHT appear, as far as I know there are certain faith requirements for the tombstone to appear (in the same place where she was standing between Majula and Heide's Tower.

A link to the complete information here.