Let me start by being crystal clear about one thing: Do not drop your existing whole life policy under any circumstances. No matter what you're paying, with a health issue such as your cancer diagnosis, your policy is a bargain! Even if you reach a point where you feel you can't afford the premiums anymore, don't give up on your coverage. All whole life insurance offers many different options for keeping your policy in force.
If you're facing a financial hardship that's only temporary (for example, if you're between jobs), you can use some of the cash value to help pay the premiums for a while. If the hardship is long term, the policy has an option that will keep the death benefit the same and let the cash value pay the premiums for years, until the cash value runs out. This is known in the insurance industry as "extended term insurance." Consult with a life insurance professional to see what other options you have.
As for whether you can qualify for new life insurance -- especially term life insurance -- have your insurance professional search the market for you. Don't be surprised, however, if you learn that you need to have been in remission at least five or 10 years to qualify for any new life insurance policy.
Remember that if you can't buy insurance in the open market, you still can get certain types of life insurance, regardless of health. They include: mortgage life insurance through your mortgage company; group term life insurance at work; life insurance through various nonprofit groups, such as a professional organization or the Knights of Columbus; or credit life insurance on car loans or credit card balances.
I recommend that you get as many of those as you can. Good luck!