1.If your present tax credit claim is as a single parent (which it should be), then as soon as you move in with your new partner you must tell the Tax Credit Office. Your present claim will end and you must take out a new couple claim. This will be based on your partner's income as well as your own. Whether he is paying towards your children's costs is irrelevant. There will be an allowance in the claim for your own 2 children. If he only has his own children part time & their main carer (normally defined as the person getting Child Benefit) is someone else then his children can't be included in the claim.
2. If your present claim still includes your ex-partner then you must do something different immediately. If that is the case, post again & I will advise you.
3. You are unlikely to be able to do anything about the house without the agreement of your ex. Also, if it is in negative equity (which I assume from your mention of losing £22K) the mortgage lender will probably not agree to you selling it anyway. You say your ex is paying over half the mortgage, so I assume you are paying the rest. How much are each of you paying each month? How much is the house worth & how large is the outstanding mortgage?
4. If you can't afford to go on as you are, you could stop paying the mortgage. Unless your ex then paid all of it the lender would then repossess the house & sell it. You & your ex would both be jointly & severally liable for the whole of the resulting debt. You would then have to come to an arrangement with the lender to pay this off over a period - how long it would take would depend on how much you could afford to pay. Alternatively, you could dispose of the debt by going bankrupt but whether that is wise depends on your financial circumstances as a whole, & you would need advice (e.g. from local CAB or CCCS - by phone).