Hallo ProdigalSon
I would suggest that you change to the Ubuntu-Mate distribution, which is an official “flavour” of Ubuntu.
The the tutorial (step-by-step “I’m-holding-your-hand” guide) by wolfman is easy to use and accurate. Thanks wolfman. 
I’ve just used it as the basis for something I’ve been worried about for a few months now - everything went well, no problems at all. You should of course print it out before you begin and read it through before you set off.
You said you have a dual-boot set-up:
- If you have only one hard drive in your computer wolfman’s guide applies as written. Just follow it step-by-step.
- If, however, you have two hard drives in your computer you should be aware that wolfman’s guide will need a little extra information to help you find out where all the different bits are installed across the discs.
I’m not trying to get you worried, it sounds as if you have a “normal” one hard drive computer. If you do have more than one hard drive please get back to us here for more help.
I’ll just tell you two little tales from my personal experience to give you the confidence in Ubuntu to make the change.
Tale one
I started off using Ubuntu (with the standard Unity desktop) about three years ago on an old AMD 4400+ machine. The computer was indeed old and I kept having “brown-outs” where the screen would fade to almost black, nothing would happen for five to ten seconds before it came back and I could carry on. I consulted the Internet, and learned that linux with a more “lightweight” desktop might avoid this problem. Again I consulted the Internet and it seemed that Ubuntu-Mate would be suitable.
I backed up my data using the “Backup” program provided with Ubuntu (Unity) and installed Ubuntu-Mate on the old computer. I attached my backup drive, clicked on “restore” and… wow! Everything was back in the same place that it had been before! As I installed programs such as Firefox and Thunderbird their data (bookmarks, local mail folders) all “appeared” where they had been previously. This happened of course, because both distributions use the common “Ubuntu” base.
Tale two
Earlier this year I said goodbye to my over eight years old pc and bought a small noiseless computer from a company that sell Linux computers. They pre-installed Ubuntu-Mate 15.10 for me.
When the support for 15.10 came to an end it was time to upgrade. This is absolutely no problem with Ubuntu based distributions, provided that the computer has the entire Ubuntu-system installed on one single drive. My new computer had two drives; a small fast SSD for the “system” and a traditional spinning drive for the my data. I had chosen this set-up as the best compromise between cost and performance for my budget. So I knew I would have an “interesting” upgrade…
I tried the usual upgrade procedure (not really knowing any better at the time) with no success. It couldn’t locate the different parts of the system as they were spread over two discs. Back to the Internet.
I had previously read wolfman’s post and consulted it. I complemented it with a post from the “Ubuntu” forums from 2008 and some basic Linux file system information from my Linux books. I couldn’t “upgrade” my system from 15.10 to 16.04.1, I was going to have to re-install it. The first step was of course the backup(s) (I made two, one was a simple “copy” of all the data, not a proper backup).
I used wolfman’s guide as the basis and modified the parts affected by my dual-drive set-up according to my “research”. It worked perfectly at the first attempt (thanks @wolfman 
).
Your situation
My understanding of Ubuntu and Ubuntu-Mate suggests that the following should work:
- Obtain and read wolfman’s guide.
- Backup your data using the “Backup” tool provided in Ubuntu (the backup should be on an external hard drive).
- Download the correct Ubuntu-Mate ISO from the Ubuntu-Mate website. Is your computer a 32bit or a 64 bit machine?
- Prepare an installation DVD or USB-stick from the ISO.
- Change the BIOS settings on your computer to boot from the DVD or USB-stick.
- Insert your installation medium and re-boot the computer.
- Follow the instructions that appear, one of the options should be to replace the existing “Ubuntu” installation (leaving the Windows 7 partition alone). That’s the one to choose. Follow the installation instructions until the end.
- Remove the installation medium and change the BIOS settings back to boot from the internal hard drive.
- Start up the computer, enter the new Ubuntu-Mate system. The welcome screen appears, you can close it for now and come back to it later.
- Attach your backup and go to System>Control centre>Backups. Note, “Backups” is near the bottom of the “Control centre” window, you might miss it at first glance. Use “Restore” to restore your data.
- When you have restored your data remove the backup drive.
- Go to System>Welcome. I strongly suggest that you work your way through all the “possibilities” that it contains. And I hope you will enjoy using Ubuntu-Mate.