Dear Boothbay,
You probably have a bad O-ring at the spout base. I would like to know the faucet manufacturer though. They're all built different. As for the water going from cold to lukewarm, that can be a lot of things. Whoever installed your plumbing may have the hot and cold water lines touching at some point, the line may be too near the furnace or water vent, there may be a crossover valve installed and left open, etc.etc.etc. That problem is a search and find.
Good Luck,
Len

Many kitchen sinks and bathroom vanities have a heating vent under them at floor level. This warms the pipes a couple of feet below the faucet when the heat runs.
In my kitchen the pots and pans under the sink are warmer in winter than in summer, and the "cold" water is relatively warm for half a minute. Could yours be like that?

I don't have a 'heating vent' below my sink at floor level. How do I get to replace the 'O' ring, without taking the faucet completely off?

Dear Boothbay,
If you've already purchased the repair kit for your faucet, there should be detailed directions included.
Len

One suggestion, if it's not too late. I've found that for Delta faucets you can buy Delta replacement parts and you can buy other manufacturer's replacement parts for Delta faucets in the same store. The real Delta parts cost a bit more, but I believe they're better quality and fit - worth the difference.

The faucet is of Moen manufacture.