Yes
The text of the spell says that the creature must move away from the caster. If the grappler carries or drags the caster with it, there is no change in distance between the grappler and the caster.
The only scenario where I can imagine where the caster remains grappled is if the target has a speed smaller than its reach; e.g. it has a speed of 5 ft but a reach of 10 ft.
I don't know of any other RAW that would contradict this. However, I think that the spirit of the spell is that the target recoils from the caster, and as a GM I would be willing to entertain the idea that failing the save implies that a creature would release the grapple.
Weasels, in real life, lock their jaws when they bite down. That's what this is modeling - it bites, and now its jaws are locked and it's attached to you. It doesn't have to make any additional checks to stay attached, grapple or otherwise. It is "grapple-like" in that it's attached to you and loses its Dex bonus, but other things (grappling/grappled conditions, etc.) don't apply.
You can move, but it doesn't unattach the weasel. If it's just a normal sized weasel, it's probably pretty easy to move with it attached to you. If it's giant, then it's probably not so easy; you would need to do a grapple check yourself and use the "move" option. It is not technically grappling you, so they don't say you are grappled. It doesn't move into your square (unless it was a Tiny one and had to in order to attack you).
The way to unattach the jaws is use grappling yourself to achieve a pin (and unlock the jaws). Or kill it, though technically in the real world this doesn't get the weasel's jaws unlocked, so it would be entertaining to make them go through more work to get it off them.
I had a weasel familiar in one campaign but using it to kill downed opponents was about all it was safe to do; because when it bites it's only doing like 1 hp/round and can easily be beaten to death by most any opponent with its newly lowered AC.
In e.g. AD&D 2e the weasels have more description that makes it clear that they are biting down and continuing to suck blood. This means that it's clear the weasel isn't supposed to be able to attach to multiple opponents. As this question isn't tagged [rules-as-written] I see no reason to go past common sense to "well it doesn't SAY it can't do that" lengths.
If you would like to make this easier, the Pathfinder giant/dire weasel just has "grab" and then the blood drain, so it works more within the rules.
Best Answer
A controlled mount cannot grapple
A grapple is a "special melee attack", it is a part of the Attack action. A controlled PC mount, despite being a wolf, cannot attack. It has only Dash, Disengage, and Dodge actions, see PH page 198 "Mounted combat":
It can if it acts independently, but it still needs a free hand
Technically speaking, you can't give an order to grapple. But a mount can do things on its own. See PH "Controlling a mount":
The wolf still isn't eligible though, since it have to use a free hand, see PH page 195 "Grappling":
A wolf has no hands and it has no actions in its stats block, allowing it to grapple without hands. Compare that with a Giant Octopus tentacle attack, explicitly saying: