Will a lighter arrow cut through the wind better? I use heavier arrows (not really heavy) and I get a lot of erratic arrow flight when there's even a little bit of wind.

I don't think so.  If anything, a lighter arrow would be more susceptible to being buffeted by the wind.  

A smaller diameter shaft and smaller fletching, and smaller point if a broad head,  will reduce the effects of wind, but all arrows will be blown somewhat off course by a crosswind.  Higher FOC will keep the arrow on track a little better, but the tail end of the arrow will often be blown more out of line such that the arrow impacts at an angle.

I am not an expert by a long shot but I also had that problem a few weeks ago but it only happened when there was moderate to heavy wind. Same as you are having, erratic arrow flight not just the arrow hitting at an angle from a strong cross wind. I started tinkering and checking my tune and my arrows were a tad stiff. It also happened sometimes when my release wasn't perfect. I added more weight up front and they now fly great even on windy days. Try checking your tune and maybe your arrows are a little too stiff or too weak. Hope this helps.

I'm with Orion.  That is one reason why I switched back to aluminum and carbon from wood.  I always shoot judo points while walking to my stand.  I noticed that when the wind was blowing moderate to heavy, my wood arrows had poor to horrible arrow flight.  With carbon and aluminum, my tail end of the arrow was blown out of line some but the arrow still flew point on.  I had them all tuned (as good as my form will allow) but the carbon and aluminum are in the high end of FOC while the wood were just "normal".

Ive yet to find an arrow unaffected by cross wind. As stated above, higher foc, and smaller fletchings have performed the best for me. Woods, aluminiums, carbons,, are all effected if they are comparable specs other then material types.

I have found one of the biggest issues for unstable flight is the release. Poor releases are a sneaky culprit, and many other diagnosis are made to blame unknowingly. For hunting on windy days, shorten your shots, as goes with the code of archery ethics.