So I have the equation (x-3)^2 + (y-1)^2 = 5, and for some reason I cannot properly rearrange the equation to get y= so that I can put it into my graphing calculator, I managed to get a half circle at one point but it's supposed to be a full circle

Part of the problem is that the equation doesn't fit the definition of a function.
The reason is due to the problem of 2 values of y for each value of x.
your 1/2 circle is an ok function because of the single value. Can you get your calculator to graph a mirror image of the 1/2 circle?

Can you have it solve and plot 2 equations?

Can you find the radius and center point
and use polar cooridinates vs cartesian?

your user manual may offer some help on circle plots