If your singer is doing the mix, I'm guessing that your PA is pretty basic in terms of not having separate monitor mixes, extra channels, or more "out" jacks to send the signal elsewhere. If your board is sophisticated enough to have a couple of effects loops, you can sometimes fake a separate monitor for yourself. Usually the "effects" channel has enough preamp power to run a set of earbuds. Granted, if it's a wired setup, wrangling with the earbud cable can take some getting used to, but you can actually tune your own mix, having yourself more present, less guitar, vocals, etc. or whatever you want. It was a $20 solution. You could do a bluetooth thing for maybe $50, too. If you want to invest a bit more, get yourself a wireless rig, because then you'll have your own volume control that will show up in the main mix as well. Small powered monitors are cool, too. You can get a 6" bluetooth self-powered studio monitor for less than $200, and set it up somewhere near your ears. As long as you don't create feedback, you will hear yourself just fine. Lastly, I made something out of plexiglass once that helped a lot. It was a circle (could be square, too) of about 6-8", with a hole cut into the middle. I used a toilet tank rubber gasket to line the inside of the hole, and an SM 58 fit PERFECTLY in there. Then I clamped it to the sax bell, and angled the plexi so it reflected the sound from the bell right back towards my ears. It also tends to reflect the higher frequencies as well, which is what you actually want to hear in the monitors, as regards intonation, tone, etc. Your partials are going to get eaten up by the bass and drums, so learn to put up with a "tinny" sound, because it'll be much more accurate once it makes it out of the main speakers.
That doesn't address your issues as far as the audience hearing you, though. A singer should KNOW that he/she cannot hear themselves properly while singing, due to the physical mechanics of the throat, ears and larynx. This is true for every human being on planet Earth. So he can hear you when he's not singing, and it seems loud to him, then when he's singing, he can't hear himself properly. Add a pinch of narcissistic ego, a few drops of alcohol, and you've got trouble.
I always play at about 60% of my actual volume potential during soundcheck, so I have some "headroom" when the band kicks in. I also use a setup with a LOT of opening in the mp tip, and a horn with an open sound, like MK VI, Yamaha, Yani, etc. Some rock players swear by Dukoff, because it's very powerful and bright, but I could never get more than a squawk out of that mp, no matter what the setup. (but that's just me.)
You may have to politic a bit if you want to be heard. Enroll the other band members to listen and give their opinions on whether you're too loud, too soft, etc. and then find ways to gently gang up on the singer to get things balanced out. If he doesn't get confronted heavily, he might get that the band needs a different mix for the sax to be heard. If he won't listen to anyone but himself, then I suggest either you or he needs to be replaced because he's utterly unprofessional and toxic to the future of the band. It always seems to be the singers and lead guitarists. God help you if he does BOTH.