Your question points to a concern brought up by many. When people refer to gift amounts of $10,000 per year - that amount is based on a reference to a tax provision. This tax provision is the amount allowed by IRS to be given per year to as many people as you would like without gift tax. Often referred to as the "annual exclusion". The current amount allowed by IRS is $13,000.
These amounts have no connection to the medicaid laws that you reference for nursing home care. Federal law has changed the gifting provisions for medicaid and each state has various implementations of the Federal law. In Florida - even nominal gifts of $100 can be a disqualifying transfer. You will need to seek specific advice from NJ counsel but I can safely say that all gifts during the look back period (currently 5 years from application for medicaid) have an impact and can cause disqualification.
As a New Jersey attorney, I will agree with my FL friend above.
The two concepts are completely independent. You can gift up to the gift tax exemption amounts without it impacting your total gift tax exemption during your lifetime. (Did that sound confusing? Of course it is.) The IRS tells you that you can make gifts up to a certain amount during your lifetime, but the only gifts that count against that amount are non-exempt gifts. By contrast, the $12,000 (or whatever it is in a given year) is an exempt gift.
On the other hand, EVERY SINGLE GIFT YOU GIVE during the applicable look-back period is reportable and can have an impact on qualifying for Medicaid. There are some things you can do in planning that would not be a reportable gift and options are available, but the sooner you begin, the better.
By the way, to complicate things further, if you are in an assisted living facility (or a nursing home for that matter), and you are a veteran or the surviving spouse of a veteran, you are eligible for Aid & Attendance benefits. You don't have to have been wounded in combat for these benefits, but you have to meet certain asset qualifications. The short of it is, that for these purpose, no transfers count against the qualification. The amount doesn't matter, neither does the timing.
However, a comprehensive estate plan will look into all of these things and help you put in place a plan to keep as many options available (and protects as much money as possible).