Theresa, Just got involved here but if it's not too late, the problem you're experiencing is FLOW. Water is all about flow, how easy or how difficult. One of many ways Plumbers keep this in mind is when they try to use fewer 90 degree elbows in the run of the pipe, more 90s increase friction and lowers the water pressure down the line. This, essentially, is what is happening to you. When your husband turns on the hot water faucet at the basin the flow through that tap is much less restricted than the flow allowed through the shower hot water tap. Really, it may be easier to think of the shower as 'pushing back' harder than the basin tap does. The water(cold or hot) will flow greater through the less resistant path. The temperature at the shower drops because the Hot water is being 'pushed back' harder by the shower tap. The effect is that the turnover of hot water in that pipe is greatly reduced because the flow essentially stops(or at least becomes very slow) What you want is *Balance* between these fixtures. A temperature regulator will balance the flow of hot versus cold but you may find that the effect is very low pressure in both hot and cold shower taps in order to achieve the balance. Fixing the flow resistance to the shower is your best long term fix. A cheap fix and a way to test this for yourself is to add a restrictor to the hot water tap of the basin. This will act to balance the relative un-restriction of the basin tap with the restriction of the shower tap. And, by the way, that moan in your shower tap is another indication that flow through the tap is altered in a way that the manufacturer didn't design in. As a quick test simply turn off the hot water feed under the basin, turn it back on slowly until the hot water in the shower is unaffected by the basin flow. You can then buy flow restrictors to install in the feed lines under the basin to make a more permanent 'fix' or if you can live with it you might just leave it that way. By far though, your best long term investment for your comfort and safety is to do the right job and that's to replumb. The shower is an indication of potential other problems, maybe it's only bad design in the shower tap but you probably want a more definitive answer to whether that is the case or not.
If you replumb it yourself, be sure to inspect the run of piping all the way back to the hot water heater and cold feed for each appliance on that circuit. This includes the basin, shower and toilet. You're looking for less than straight runs to one appliance versus another that would either introduce or allow less restricted or greater restriction to one appliance over another. It needs to be done for both hot and cold lines because it is the balance between hot and cold that affects your comfort, greater or lesser pressure may be a nuisance but greater or lesser temperature can be dangerous. Sorry about the length but the info should help guide you to do the right thing which is always of course what makes sense to you. I suggest doing the big job over the little, shortcuts are never good money. Good Luck, K. W.