Disclaimer - info below only applies to Massachusetts banks. I did not contact any California banks, so I have no idea if the laws or practices there are different.
The only way to keep a single mortgage and get cash in your pocket is to refinance. This may or may not work for you - depends on your current loan to value ratio (i.e. how much equity do you currently have in your house), and your present interest rate. We found that banks will usually charge a higher rate if you’re “cashing out” than you would if you just refi’d your existing principal balance.
A home equity loan is just a vanilla second mortgage. They agree to lend you money at a fixed or variable rate, and you get it all at once, and can do anything you damn well please with it. If you do spend it on home improvements, you can deduct the interest from your taxes. Since you’re getting all the money up front at once, you need to decide the exact amount you need so you don’t over or under borrow.
A home improvement loan is pretty much just a restricted home equity loan. Some places will give you a slightly lower rate, in return for control of the money - they agree to lend you $50k for an addition, but you don’t get any of the money until the work is in progress. You have a plan to follow, and they send out an inspector when you want a release of the funds - say you get 25% when the framing is done, another 25% when the drywall is up, etc. These are more usual for new construction loans, but we did find a bank or 2 that did it for additions. The bank offers a lower interest rate than they would for a standard home equity loan, because they’re confident the value of the home will increase as they lend you money - you don’t get cash unless their inspector is satisfied that work is being performed.
A home equity line of credit is a loan where you make withdrawals as you go, up to a fixed ceiling. So you can withdraw as you go during your project, and only pay interest on the amount you’ve actually withdrawn so far. All of these we found were variable rate, fixed for 2-5 years at first. Also, the ones we found had 2 periods - a 10 year draw period, followed by a 10 year payback period. During the draw period, you can withdraw money as often as you like, up to the ceiling. You only have to pay interest on the loan, not principal, though that’s a fools game. After 10 years are up, you can no longer withdraw, and any principal balance left converts to a 10 year variable rate mortgage.
Having reread your OP - new carpets and floors probably isn’t something you want to borrow money over. It won’t increase the value of your house, the way an addition of finished basement would.