I was in exactly the same situation as you, and yes, you can totally get Taiwanese nationality without giving up your U.S. passport.
However, you’ll need a whole lot of paperwork and you’ll have to spend quite a lot of time in Taiwan. It’s also very likely that you will need your mother’s help for a lot of the process, so start reminding how much you love her now. 
Here’s how it goes:
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GET AN OVERSEAS PASSPORT. You must do this at the TECO that services the area where you come from in the States. You will need your mother’s passport (with ID number), your parents’ marriage certificate and your birth certificate. (The American documents will need to be authenticated by the TECO that services the area WHERE THE DOCUMENTS WERE ISSUED.) If your mother doesn’t have a valid Taiwanese passport, she must get one. (This is totally possible, even if it’s been decades since she’s had one. The Taiwanese will always recognize its people, even if they take other nationalities. Mom rolled up with the passport she came to the U.S. on when she was 17 and they still renewed it. She did have to provide extra documentation though, including an official Taiwanese address.)
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GET AN ENTRY/EXIT PERMIT. Once you have your overseas passport, you have to get an entry/exit permit pasted into it. This allows you to enter Taiwan for up to 90 days. You get this at TECO and I think you can apply for it at the same time you apply for your passport.
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COME TO TAIWAN. You must enter with this new overseas passport with valid entry/exit permit. DO NOT ENTER ON YOUR U.S. PASSPORT.
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APPLY FOR A TARC. Once you arrive in Taiwan, you must apply for a TARC, which is a residence card for overseas Taiwanese (and others awaiting nationality). DO THIS THE MOMENT YOU ARRIVE. You will need all kinds of documents, including a background check (from your home country), a health check (done locally), your mother’s household registration book (so you’ll need to find a friend or relative to help provide this if your mom doesn’t have property in Taiwan anymore), your mother’s passport, your mother’s ID card (she might have to come to Taiwan and get a new one, which is a simple process, but might require her physical presence), your overseas Taiwanese passport, your U.S. passport (I think), your authenticated birth certificate, your parent’s authenticated marriage certificate (even if they are divorced now), and some passport photos. Also, your mom might have to register her marriage to your dad if she has never done so. This means he will need a Chinese name, so you might have to find one for him. (Your Taiwanese ID will list your parents, so this is why you might have to do this. In my case, my father passed away before I applied for nationality, and my mother never registered him as part of her household, so there’s a gap on my ID where my dad’s name should go.) IF YOUR MOTHER COMES TO TAIWAN TO HELP WITH THIS, BE SURE SHE ENTERS WITH HER TAIWAN PASSPORT.
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WAIT A LONG TIME. The 360 day requirement starts ONLY AFTER THE TARC IS ISSUED. I stayed in Taiwan the full year without leaving, which is the fastest way to meet the stay requirement. You can also do 270 days each year for two years, or 183 days each year over five years. But remember, THE CLOCK DOESN’T START TICKING UNTIL YOU HAVE THAT TARC, so make getting it a priority the moment you arrive.
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PROVE YOU HAVE WAITED A LONG TIME. Once you have met the residency requirement, go back to immigration and get a paper which confirms that you have stayed in Taiwan long enough. (You just give them your passport and TARC and maybe a SASE which you can buy there.) This takes a couple of weeks to process, I believe. Also, I’m pretty sure this is the last time I saw my TARC, so make copies of it before you hand it in because it’s likely you’ll need them later.
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GET ANOTHER HEALTH CHECK. I can’t remember if you need this before or after the proof of stay part, but you definitely need it before the next step.
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REGISTER AND GET AN ID CARD. After you have this paper which proves you have stayed in Taiwan long enough, you can register yourself at your housing office. You will need a housing registration book (your mother’s, or one of your own if you know a landowner who will let you use theirs), this proof of stay paper, the health check (I’m pretty sure), and a photo. This whole process takes about 10 minutes and you’ll walk out with your new ID card and nationality.
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GET A NEW PASSPORT. Now that you have an ID card (with number) you must apply for a new Taiwanese passport on which you can now exit Taiwan.
This whole process takes a whole lot of time and energy, and there will be all kinds of unexpected setbacks (seriously, even on the day I applied for my ID card, the photo machine broke). However, you’ll find that while the procedures are mind-numbingly detailed and ridiculous, Taiwan is really happy to have you. No one is going to bend the rules for you, but you’ll see that the many, many bureaucrats you will have to deal with in this ordeal really DO want to help you.
My whole process took about a year and four months, including a bit of lag time getting the TARC, then a bit of feet dragging on my part at the end with the health check and ding ju. Also, I think I had to wait about a week for my new passport. However, I’m really glad I went through the process and it’s awesome being a dual-national now!
Also, some notes:
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Even though more and more of us are coming back now, MOST PEOPLE have never seen an overseas passport and a lot of people don’t know what a TARC is. I’ve had confusion at banks and phone companies, and the HR at my old job had to learn a few things about processing my papers.
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MAKE A LIFE FOLDER. You never know when you might need which document, so I made a big folder with all of my papers in it and just took it with me anytime I had to do anything administrative. My mom also left me her ID card and chop when she returned to the States and this was also handy to have when I had to navigate this process without her.
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KEEP COPIES OF YOUR TARC EVEN AFTER YOU GET YOUR ID CARD. You’ll find that you’ll have to update a lot of things (bank accounts, phone accounts, health insurance cards, etc.) once you have your new ID number and these places will only have your old ID on file. Having a copy of the old card is super useful.
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YOU CAN GET A WORK PERMIT. My first company had to apply for my work permit (which you will still need to take employment, and for reasons I don’t understand, they need your U.S. passport to get it), but I learned when I went part-time that I could get my own work permit after getting the TARC. This permit allows you to work anywhere and you won’t have to be tied to a particular employer.