The only limitations in terms of the distance you are able to travel with this are your chakra and the ice that is present on the ground, as it is a channeled technique.
While you are free to turn and curve your path as much as you'd like, you're not going to be able to make sudden and instantaneous turns in the manner that you're describing. If you want to suddenly turn to the left, the path of motion would contain rounded angles rather than sharp ones, if that makes sense. Think of it a bit like actual ice skating, except a lot easier to control.
While you can use this while your paralyzed, there's a good chance that the sudden movement is just going to end up knocking you over unless you were paralyzed in a stance with an appropriate weight distribution for this type of movement.
Unless you have any more questions,
(Hyouton: Yuki Tenkan)-Ice Release: Snow Conversion
Type: Supplementary
Rank: B-Rank
Range: Long
Chakra: 20
Damage: N/A
Description: The user manipulates the snow on the battlefield that they created and uses their chakra to completely freeze the snow, causing it to become ice. Upon turning into ice, the newly turned ice will retain the same properties that it had before becoming ice. This includes, shape, rank, momentum, location, etc. Due to the small amounts of air in the snow and the increased density of ice, there will be slightly less ice after the snow is transformed.
Note: Can only be used four times per battle.
Analyze or something