I know you can go to Riften but I don't want to do that just yet, and I don't have the cash. I accidentally screwed up my character wight and I don't want to restart. Is there a way to use the console to change the weight? If so, is it safe to do without causing any problems?

Just go to the lady in Riften if you already have the Dawnguard DLC, I do not see a reason why you can't

But if you are low on money, sell stuff or use this console command:

You have to options, first one which is fast and easy. is use showracemenu command. no negative effects will happen as long as you dont change the race. the eyeliner might disappear and the skin tone might get a bit dark but you can easily put them back using the sliders. another option is the one tgl mentioned. type player.additem f [gold]. then you can remove what you gave to yourself by typing player.removeitem f [ amount of gold].