Believing that fixed-price protects you from time or cost overruns is a common misconception, but is completely untrue. There are four reasons for this.
The reality is that you do not know exactly what you want. If you go fixed-price with anyone competent they will then need a change control process, and every change you introduce will be costed and will add time. The final cost and time will not be what you started with.
Things take as long as they take. Signing a contract doesn't magically make it quicker.
There is only one way a supplier can safely offer a fixed price on a high-risk project (all new product builds are high risk from a supplier POV) and that is by pricing in the risk. Want to pay 50% more? Go fixed price.
The thing that you actually care about but assume will be present. It will not necessarily be present. If a supplier is working fixed price and the deadline looms with a lot of work to do, they are likely to rush. If they rush they will produce dogfood.
If you launch your shiny new product with dogfood it will fail.
Once you have a big pile of shit code it is virtually impossible to remediate. I speak from bitter bitter experience here. You may well have to throw it away and start again.
You really really actually care about quality, assume it will be present, often it isn't, and it is one of the hardest things for you to control.
Work on an incentive structure with your suppliers that helps you deliver on your real requirements - which include change and quality.