Definitely a much larger pot. It sounds like you are doing all of the right things. But it needs more room for big healthy roots.

Also -if you are hand watering it as opposed to letting the rain do it for you try to keep the water from the hose off the leaves.

I guess it was actually my great grandmas so maybe over a 100 years old.

Can you explain in more detail “Just trim off about an inch from the ends of the roots to stimulate new growth.” I’m not following exactly what you mean.

Many old roots tends to dry out at the ends. So to stimulate them into new growth, you just cut of about an inch from all the ends of the main ones. Don't bother with the hairlike ones, just the ones that look like shoe laces and any others that are up to about half inch thick. Leave the other roots that are thicker than a half inch, because of the rose's age, they are best left alone. The larger pot is the main item to keep the rose healthy. Roses push their roots downward and don't like to be cramped or hot. They prefer damp, cool roots. A larger pot will be cooler. Just to nitpik. If you have a choice of a pottery pot or a fibreglass one, take the fibreglass. Ceramic pots can get very hot in high heat and damage rose roots. Fibreglass ones stay nice and cool. Also much easier to handle as they are light.

Re: "The problem is that when it grows it grows these long shoots that just
end up flopping over. I can’t get very much upward growth it seems." You may want to back the new pot up to some sort of trellis, and tie the long shoots to it to support them, to see if it's happy with that support. For all you know, you may have a climber.