I have a few thoughts - take it for what it's worth...
Note: I was a nanny in college to two older, full time working parents, and have three kids of my own now.
1. Sounds like your daughter IS demanding your attention at home because you are both full time working parents...i.e. she doesn't get your attention during the day. Make sure that you are spending some quality time with her in the evenings and on the weekends where it's just about her and what she wants to do. My kids LOVE "Daddy time" or "Mommy time" even if it's helping Dad fix stuff around the house and run to Home Depot.
2. Read the book Love and Logic - quick summary, HER choices lead to consequences. You are not a "mean parent" you are not "making her mad" etc. SHE has the ability to make choices that affect the outcome. You, as her parent, will enforce consequence to her behaviors, both positive and negative. Yes, a little "deep" for a 3 1/2 yr old, but start NOW.
3. You and your husband need to have a united front - NEVER argue or change answers in front of her. Discuss things after, out of her earshot. Whoever starts handling her, ENDS handling her. Even if you disagree with how your husband is handling a situation, support him, follow through with his approach until the issue is over. This teaches your daughter that her parents are a team and she can't change what they say or do.
4. Make a little, easy chart that lists "consequences" for some of the biggest issues - talking back, getting out of bed, tantrums, etc. Follow through on them EVERY TIME. Tell her, "I'm sorry you made the choice to start screaming and throwing a tantrum, that means that you will be going to bed 3 minutes earlier tonight or you have to sit on the step for 3 minutes," whatever you decide the consequence is. Have as many REWARDS for good behavior too! Kids this age love sticker charts.
5. Do NOT have discussions with a 3 year old DURING an argument. It IS because you asked her to. Enough said. She can think whatever she wants, but she still MUST DO what you asked. If not, she made the choice to not listen...consequence!
6. Make sure that after dinner, the house gets darker, quieter, tv is off, etc. We have a rule that when PJs are on, the whole house settles down. This is "relaxing time". Snuggle up and read with her on the couch, talk about her day (highs and lows), then go brush teeth and go potty and tuck her in. THAT'S IT. Repeat EVERY night. Kids thrive on routine.
7. Make sure she's going to bed no later than 7:30-8pm. Kids get overstimulated very easily. Then it's super hard to get them to stay down.
8. If she gets out of bed/her room/screams I like the "super nanny" approach of "Good night sweetheart, it's time for bed" as your walking her back into her room. Tuck her in and leave. The next time she comes out of her room, no talking, no attention, walk her back to her room and leave. Do this 100 times if necessary - IT DOES WORK! It teaches her that no matter what she does, she will not get anymore attention and she will give up. You must stick with it EVERY time though. No yelling at her, no discussions or laying down with her or letting her into your bed - all you're teaching her then is that she CAN control this situation and get what she wants. (Note: She doesn't have to be in her bed, but she does have to stay in her room. Who cares if she falls asleep on the floor?)
My kids know, "Mom says 'Goodnight' and that's IT." I give them plenty of love, attention and hugs before bedtime. After bedtime, it's MY time. I do not play games, I do not give extra cups of water, etc. Of course there are always exceptions, like when someone's sick, but generally speaking, I do not tolerate "drama" at bedtime. For us, the consequence is earlier bedtime the next night, which usually also means missing out on some bedtime story time too then. I remind them, it's your behavior last night that means you have to go to bed earlier tonight.
Let us know how it goes!!!