I have an old tub with a build a shower kit, when I pull the diverter for shower mode the water still comes out the spigot?

You should be able to simply unscrew the old spigot/diverter and replace it with a new one... replacement diverters are sold at any home improvement store and are relatively cheap. Depending on the age of the plumbing, you should have no issues with a universal faucet. I had to buy a custom adapter for mine, because it was original plumbing and the thread patterns have changed sometime within the last 50 years...

I have an old tub with a build a shower kit, when i pull the diverter for shower mode the water still comes out the spigot?

Although you didn't say I'm assuming that you have a three handle valve with the diverter in the center.(see image) Shut the water off and remove the diverter stem and washer assembly. Check and replace then washer if necessary. Check the seat for nicks and rough spots. If you find any replace the seat also. Now turn the water back on amd turn on a side to flush out the valve. Reassemble and test, good luck, Tom