Tech support mispoke (to be gracious). Garmin Connect is the website where you view your data. It has not gone away (got a major update a few months ago). Garmin Express is a software for communicating between the computer, GPS unit, and Garmin Connect; it's the New Way. The ANT+ Agent was the old way and still works in many cases, and is worth trying if Garmin Express is giving you trouble, particularly since the 610 shipped well before Garmin Express came along.
You can try a soft reset first: hold down on the lap button while turning on the watch with the power button (hold lap, tap power), select Yes at the prompt. This will remove settings but not your saved activities. If you've managed to get all your activities off it, it may be worth trying a
master reset, which will delete everything.
Unfortunately DC Rainmaker doesn't have a forum, but people having problems with a particular product often post in the
review for that product. Might be worth a shot, and good luck.
Is this a different question, or were these issues mixed together in your original post? Sometimes the updates don't go through properly, or two updates come out together quickly. I wouldn't worry too much about two updates close together and would go ahead and let it do the update...if it then keeps saying the upgrade still needs to be applied, then I'd worry. I'm still on a 76CSx for hiking GPS, so don't know much about the new stuff.